# Cosmic Birefringence as a probe of dark matter nature: Sterile neutrino and
dipolar dark matter

Jafar Khodagholizadeh<EMAIL_ADDRESS>Farhangian University, P.O. Box
11876-13311, Tehran, Iran  S. Mahmoudi<EMAIL_ADDRESS>Department of
Physics, School of Science, Shiraz University, Shiraz 71454, Iran  Rohoollah
Mohammadi<EMAIL_ADDRESS>Iranian National Museum of Science and Technology
(INMOST), PO BOX: 11369-14611, Tehran, Iran,
School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P.
O. Box 19395-5531, Tehran, Iran  Mahdi Sadegh<EMAIL_ADDRESS>School of
Particles and Accelerators, Institute for Research in Fundamental Sciences
(IPM), P. O. Box 19395-5531, Tehran, Iran.

###### Abstract

Recently, non-zero rotation angle $\beta=0.30^{\circ}\pm 0.11^{\circ}$
$(68\%\text{ C.L.})$ [Phys. Rev. Lett. 128, no.9, 091302 (2022)] has been
reported for linear polarization of cosmic microwave background (CMB)
radiation, which is known as cosmic birefringence (CB). We used this
birefringence angle of CMB to study and distinguish different candidates of
dark matter (DM), e.g., dipolar and sterile neutrino DM. We calculated CMB
forward scattering by those probable candidates of DM to generate $\beta$ in
the presence of primordial scalar fluctuations’ background. We explicitly
plotted bounds on the mass and electromagnetic coupling for different sectors
of DM, sterile neutrino, and dipolar DM, and compared them with other
experimental bounds. Regarding dipolar DM, our calculations put a bound on the
Majorana magnetic dipole moment about $\mathcal{M}\leqslant 1.4\times
10^{-14}\,\frac{\beta}{0.30^{\circ}}\sqrt{\frac{m_{\text{\tiny{DM}}}}{1\,GeV}}\,e.\text{\,cm}$.
In the case of sterile neutrino DM, the bound on the mass and mixing angle was
estimated at $\theta^{2}\leqslant
3.3\,(rad)^{2}\frac{\beta}{0.30^{\circ}}\,\frac{m_{DM}}{\rm{KeV}}$, which can
be a new constraint for sterile neutrino DM whose production mechanism is
motivated by models with a hidden sector coupled to the sterile neutrino.
Based on our results, if the constraint on the mass and the electromagnetic
coupling for DM must be within the allowed region, none of the considered
candidates can compensate for all the observed CB angle. We also discussed the
maximum contribution of the CB angle via CMB forward scattering by different
sectors of the dark matter.

##  I Introduction

Astrophysical observations of the past several decades confirm that the
majority of matter in the universe consists of DM rotation ; White:1977jf ;
WMAP:2003elm ; Clowe:2003tk . Up to date, several attempts have been made to
identify the physical properties of the missing mass, including accelerator-
based techniques Cai:2017mow ; Baer:2012uy ; Sigurdson:2004zp ; Kumar:2015wya
, direct detection Catena:2018uae ; Graham:2013gfa ; Duerr:2015aka , indirect
search experiments Arguelles:2012cf ; Foster:2021fxa ; Sigurdson:2004zp ;
Ramos:2021omo , and astroparticle approaches such as studying their possible
effects on the CMB radiation Mahmoudi:2018lll ; ModaresVamegh:2019nja ;
Haghighat:2019rht . However, despite significant endeavors to determine its
nature and properties, the identity of DM is still shrouded in mystery in
astronomy and particle physics Green:2021jrr ; Cooley:2021rws ;
Slatyer:2021qgc .

It seems that DM consists of non-relativistic particles that mainly interact
gravitationally. In addition to gravitational interactions, they may
experience very weak interaction with the standard model. Regarding the
electromagnetic interaction, their coupling to photons is believed to be
either non-existent or feeble. Nevertheless, an anomalous coupling with
photons has been proposed for some candidates of DM which causes various
phenomena in astrophysics and cosmology, such as superradiance around rotating
black holes Arvanitaki:2014wva ; Cardoso:2018tly ; Stott:2018opm , X- or
$\gamma$-ray emission from the decay of DM Hofmann:2019ihc ; Chiang:2014xra ;
Cadamuro:2011fd ; Arias:2012az ; Higaki:2014qua , and cosmic birefringence
(CB) Pospelov:2008gg ; Fedderke:2019ajk ; Agrawal:2019lkr .

The CB angle can be generated in parity violating interaction Lue:1998mq ;
Fujita:2020ecn ; Liu:2006uh . This angle measures the possible rotation of the
linear polarization plane for photons propagating over large distances in the
universe. Among all cosmic rays, CMB photons are an ideal target to probe this
effect for the following reasons: They are emitted at the epoch of
recombination, accurate prediction of their polarization angular power spectra
by the $\Lambda$ cold DM ($\Lambda$CDM) model is possible, and the CMB
polarization is sensitive to parity-violating phenomena. Indeed, it is
conventional to decompose the observed pattern of CMB polarization into
eigenstates of parity referred to as E and B modes whose correlations, which
are zero in the standard scenario of cosmology due to parity conservation,
lead to CMB birefringence.

Generally, CB can be considered a probe of physics beyond the standard model
of cosmology and elementary particles which breaks Lorentz and CPT symmetry.
For instance, it provides a tantalizing hint for new physics of axions. The
interaction between CMB photons and axion-like particles (ALPs) with various
potentials occurring during or after the recombination epoch could be
accounted for isotropic CB. For the quadratic and cosine potentials, the lower
bounds on some physical properties of APL have been obtained using the
observational value of CB Fujita:2020ecn . An axion with a linear potential
that plays the role of dark energy has been studied in Gasparotto:2022uqo ,
where an upper bound has been placed on the axion decay constant using the
birefringence measurement and the constraint on the equation of state for dark
energy by Planck 2018 Minami:2020odp  and the axion-photon coupling constant
by $Chandra$.

There are several independent approaches for extracting the birefringence
angle; one such approach on which most discussions of birefringence have
focused is using $EE$, $BB$ and $EB$ power spectra Minami:2020odp ;
WMAP:2008lyn ; Planck:2015qep ; Namikawa:2020ffr ; ACT:2020frw ;
Komatsu:2022nvu . However, this approach has a fundamental problem. Since this
effect degenerates with an artificial rotation of polarization angles
generated by orientation miscalibration of polarimeters, it is not possible to
distinguish birefringence from the systematic uncertainty of a miscalibration
of the orientation angle. The galactic foreground emission is one way to deal
with this problem Minami:2020fin . To this end, given that the miscalibration
of detector orientation changes the foreground and the CMB spectra in the same
way, one could use data at different frequencies to separate birefringence
from foreground and calibration effects. Another independent method has been
proposed by authors in Abghari:2022bet , based on which the birefringence
angle is determined only by using CMB temperature, $E$ modes, and their cross-
correlation. In addition to the above methods, it has recently been found that
parity violating forward scattering of the CMB in the presence of scalar
perturbation can lead to the CB effect. For instance, the authors in
Mohammadi:2021xoh  have shown that the weak interaction of the CMB and cosmic
neutrino background, in the order of one loop forward scattering, can generate
non-vanishing CB in the presence of scalar perturbation whose value is at
least one order larger than the CB angle reported by using Planck data
release. Therefore, other mechanisms resulting in non-vanishing parity
violating forward scattering should be taken into consideration to produce the
CB Khodagholizadeh:2014nfa ; Mohammadi:2016bxl ; Khodagholizadeh:2019het ;
Tizchang:2016vef ; 2022dipole .

Following the last mentioned approach, here, we introduce the two new sources
of CB, i.e., dipolar DM and sterile neutrino DM. We show that the
birefringence limit from CMB observations enables us to constrain their
coupling to photons. In other words, the CB effect provides a new tool to
investigate the DM properties and open up a new observational window to
explore the nature of DM.

The rest of the paper is organized as follows: We present a brief review of
relativistic Boltzmann equations in section II. In section III, we give a
general discussion of CB. Then, we choose dipolar DM and sterile neutrino DM
as two kinds of DM and study their interaction effects on the linear
polarization of the CMB which result in CB in section IV. In section V, we
present the summary and conclusion.

##  II Relativistic Boltzmann Equations

The linear and circular polarizations of an ensemble of photons described by
Stokes parameters are the density matrix components in the polarization space
as follows:

$\displaystyle\hat{\rho}_{ij}\equiv\frac{1}{2}\left(\begin{matrix}I+Q&U-iV\\\
U+iV&I-Q\\\ \end{matrix}\right),$ (1)

where $I$ is the total intensity of radiation, $U$, $Q$, and $V$ describe the
polarization intensity of photons and, for unpolarized photons, $Q=U=V=0$. The
linear polarization of the photon is defined in terms of the Stokes parameters
$Q$ and $U$, and parameter $V$ indicates the net circular polarization or the
difference between left- and right-circular polarization intensities.

The time evolution of the density matrix components $\rho_{ij}({\bf{k}})$ is
given by the quantum Boltzmann equation as kosowsky1996cosmic

$\displaystyle(2\pi)^{3}\delta^{3}(0)(2k^{0})\dfrac{d}{dt}\rho_{ij}({\bf{k}})=i\langle[H_{I}^{0}(t);D_{ij}^{0}({\bf{k}})]\rangle-\dfrac{1}{2}\int
dt\langle[H_{I}^{0}(t);[H_{I}^{0}(t);D_{ij}^{0}({\bf{k}})]]\rangle,$ (2)

where $H_{I}^{0}(t)$ is the first order of the interacting Hamiltonian

$H_{I}^{0}(t)=-\frac{i}{2}\int\limits_{-\infty}^{\infty}\,dt^{\prime}T\\{H(t),H(t^{\prime})\\},$
(3)

in which $T$ signifies a time-ordered product and $H$ denotes the interaction
Hamiltonian which relates to the interaction Hamiltonian density
$\mathcal{H}(x)$ as follows

$H(t)=\int d^{3}{\bf{x}}\,\mathcal{H}(x).$ (4)

Moreover, $D_{ij}({\bf{k}})\equiv a_{i}^{\dagger}({\bf{k}})a_{i}({\bf{k}})$ is
the photon number operator. The first term on the right-hand side of Eq. (2)
is called the forward scattering term which is proportional to the amplitude
of the scattering, whereas the second one is known as the higher order
collision term, giving the scattering cross section which is highly
subdominant compared to the first term.

Although the Stokes parameters $I$ and $V$ are independent of the reference
frame, $Q$ and $U$ are frame-dependent parameters. However, by introducing a
set of linear combinations of polarization parameters $Q$ and $U$ as
$\Delta^{\pm}_{\text{P}}=Q\pm iU$, one can find the reference frame-
independent parameters, where $P$ stands for polarization. Accounting for the
scalar mode perturbation of the metric and using the Thomson scattering of CMB
photons by cosmic electrons, the time evolution of the CMB radiation transfer
is governed by the following set of equations Bond:1984fp :

$\displaystyle\dfrac{d}{d\eta}\Delta^{\text{S}}_{\text{I}}+iK\mu\Delta^{\text{S}}_{\text{I}}+4[\dot{\varPsi}-iK\mu\Phi]=\dot{\tau_{\text{e}}}\left[-\Delta^{\text{S}}_{\text{I}}+\Delta^{\text{S}}_{\text{I}_{\circ}}+i\mu
v_{b}+\dfrac{1}{2}P_{2}(\mu)\Pi\right],$ (5)
$\displaystyle\dfrac{d}{d\eta}\Delta^{\pm\text{S}}_{\text{P}}+iK\mu\Delta^{\pm\text{S}}_{\text{P}}=\dot{\tau_{\text{e}}}\left[-\Delta^{\pm\text{S}}_{\text{P}}-\dfrac{1}{2}[1-P_{2}(\mu)]\Pi\right],$
(6)

where $\Psi$ and $\Phi$ are the metric perturbations, $v_{b}$ is the baryon
bulk velocity, and

$\Pi\equiv\Delta_{I2}^{(\rm S)}+\Delta_{P2}^{(\rm S)}+\Delta_{P0}^{(\rm S)}$
(7)

The superscript “S” denotes the scalar metric perturbations, and the CMB
radiation transfer is described by the multipole moments of temperature
$(\text{I})$ and polarization $(\text{P})$ Zaldarriaga:1996xe ; Seljak:1996is

$\Delta^{\text{S}}_{\text{I,P}}(\eta,K,\mu)=\sum_{l=0}^{\infty}(2l+1)(-i)^{l}\Delta^{\text{S}}_{\text{I,P}_{l}}(\eta,K)P_{l}(\mu),$
(8)

where $P_{l}(\mu)$ is the Legendre polynomial of rank $l$,
$\mu=\hat{\bf{n}}.\hat{K}=\cos\theta$, and $\theta$ is the angle between the
CMB photon direction $\hat{\bf{n}}=\frac{\bf{k}}{|\bf{k}|}$ and the wave
vectors $K$ of the Fourier modes of scalar perturbations (S). Besides,
$\dot{\tau}_{e}=an_{e}x_{e}\sigma_{T}$ indicates the differential optical
depth for Thomson scattering in which $\sigma_{T}$ is the Thomson cross-
section and $a(\eta)$ is the scale factor normalized to unity at present as a
function of conformal time ($\eta$). Electron density and ionization fraction
are denoted by $n_{e}$ and $x_{e}$, respectively.

To study the polarization features of the CMB photons in the context of
cosmology, it is common to separate the CMB polarization
$\Delta^{\pm\text{S}}_{\text{P}}(\eta,K,\mu)$ into a curl-free part (E mode)
and a divergence-free part (B mode) as follows:

$\displaystyle\Delta^{\text{S}}_{\text{E}}(\eta_{0},K,\mu)$
$\displaystyle\equiv$
$\displaystyle-\dfrac{1}{2}\left[\bar{\eth}^{2}\,\Delta^{+\text{S}}_{\text{P}}(\eta_{0},K,\mu)+\,\eth^{2}\Delta^{-\text{S}}_{\text{P}}(\eta_{0},K,\mu)\right],$
(9) $\displaystyle\Delta^{\text{S}}_{\text{B}}(\eta_{0},K,\mu)$
$\displaystyle\equiv$
$\displaystyle\dfrac{i}{2}\left[\bar{\eth}^{2}\,\Delta^{+\text{S}}_{\text{P}}(\eta_{0},K,\mu)-\eth^{2}\Delta^{-\text{S}}_{\text{P}}(\eta_{\circ},K,\mu)\right],$
(10)

in which $\eth$ and $\bar{\eth}$ are spin raising and lowering operators,
respectively. One can obtain the value of $\Delta_{E,B}^{S}(\hat{\bf n})$ at
the present time $\eta_{0}$ and in the direction $\bf\hat{n}$ by summing over
all their Fourier modes $K$ as follows Zaldarriaga:1996xe ; Seljak:1996is :

$\displaystyle\Delta_{E,B}^{S}(\hat{\bf{n}})$ $\displaystyle=$
$\displaystyle\int d^{3}{\bf K}\,\xi({\bf K})e^{\mp
2i\phi_{K,n}}\Delta_{E,B}^{S}(\eta_{0},K,\mu),$ (11)

where $\phi_{K,n}$ is the angle needed to rotate the $\bf{K}$ and
$\hat{\bf{n}}$ dependent basis to a fixed frame in the sky. Moreover, it
should be mentioned that the negative sign is used for the E mode and the
positive sign is for the B mode The random variable $\xi(\bf{K})$ used to
characterize the initial amplitude of the mode satisfies

$\langle\xi^{*}({\bf K}_{1})\xi({\bf K}_{2})\rangle=P_{S}({\bf K})\delta({\bf
K}_{1}-{\bf K}_{2}),$ (12)

where $P_{S}(K)$ is the initial power spectrum of the scalar mode perturbation
and the angle brackets $\langle...\rangle$ represent an ensemble average over
initial conditions. Finally, to characterize the statistics of the CMB
perturbations we need to calculate the power spectra which is defined as the
rotationally invariant quantity as follows

$C_{E,B}^{\ell}=\frac{1}{2\ell+1}\sum_{m}\left\langle
a^{\ast}_{E,lm}a_{B,lm}\right\rangle,$ (13)

where $a_{E,lm}$ and $a_{B,lm}$ are the expansion coefficients of
$\Delta_{E,B}^{S}(\hat{\bf n})$ in terms of spherical harmonics

$\displaystyle\Delta_{E}^{S}(\hat{{\bf n}})$ $\displaystyle=$
$\displaystyle\sum_{lm}\left[{(l+2)!\over(l-2)!}\right]^{1/2}a_{E,lm}Y_{lm}(\hat{{\bf
n}}),$ $\displaystyle\Delta_{B}^{S}(\hat{\bf n})$ $\displaystyle=$
$\displaystyle\sum_{lm}\left[{(l+2)!\over(l-2)!}\right]^{1/2}a_{B,lm}Y_{lm}(\hat{\bf
n}).$ (14)

Now, using Eq.(13) and by integrating Eqs. (11) and (12) over the initial
power spectrum of the metric perturbation, we obtain the power spectrum for
$E$\- and $B$\- modes as follows

$C_{E,B}^{\ell\,(S)}=\frac{1}{2\ell+1}\frac{(\ell-2)!}{(\ell+2)!}\int
d^{3}{\bf{K}}P_{S}({\bf{K}})\Big{|}\sum_{m}\int d\Omega\,Y^{*}_{lm}(\hat{{\bf
n}})\Delta_{E,B}^{S}(\eta_{0},K,\mu)\Big{|}^{2}.$ (15)

##  III General discussion about Cosmic Birefringence (CB)

In the context of the standard model of cosmology, due to the parity symmetry,
$E$ and $B$ mode polarizations are not correlated. However, if CMB photons
experience some sort of interaction that violates parity symmetry and the
Lorentz symmetry, the phase velocities of the right- and left-hand helicity
states of photons will differ and, therefore, the plane of linear polarization
will rotate in the sky by an angle $\beta$

$Q\pm iU\mapsto\left(Q\pm iU\right)e^{\pm 2i\beta},$ (16)

where the function $\beta$, called _birefringence angle_ , characterizes the
amplitude of deviation from the standard model. Hence, a part of $E$-modes’
polarization transfers into $B$-mode ones, and the $EB$ mode polarization will
not be zero. Such an effect could have left measurable imprints in the CMB
angular power spectra $C_{\ell}$’s. Indeed, the presence of parity violation
interaction induces a rotation of the CMB angular power spectra as follows:

$\displaystyle C_{EE,\ell}$ $\displaystyle=$
$\displaystyle\cos^{2}(2\beta)\bar{C}_{EE,\ell}+\sin^{2}(2\beta)\bar{C}_{BB,\ell},$
$\displaystyle C_{BB,\ell}$ $\displaystyle=$
$\displaystyle\cos^{2}(2\beta)\bar{C}_{BB,\ell}+\sin^{2}(2\beta)\bar{C}_{EE,\ell},$
(17) $\displaystyle C_{EB,\ell}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\sin(4\beta)(\bar{C}_{EE,\ell}-\bar{C}_{BB,\ell}),$

where $\bar{C}_{EE,\ell}$ is the standard $E$-mode power spectra, and
$\beta=0.30^{\circ}\pm 0.11^{\circ}$ ($68\%\text{C.L.}$) is the value of the
CB angle reported by using Planck data release Diego-Palazuelos:2022dsq . As
the above equations show, in the presence of parity-violating interaction,
space acts like a birefringent material, and the cross-correlators EB turn on.
In contrast, by setting $\beta=0$, one can recover the standard results of the
CMB angular power spectra.

##  IV Dark matter’s impact on the linear polarization

Regarding the possible features of DM, various experiments based on direct or
indirect methods of detection have been proposed to explore its properties.
Here, we use the CB effect as a way to indirectly examine the DM signatures.
The key point is that if the physics behind DM leads to the parity symmetry
violation assumed in the standard model of cosmology, the linear polarization
of the CMB can rotate due to its coupling to this dark sector. This results in
a non-vanishing B mode and parity-violating EB correlations. Taking into
account this point, we consider two kinds of DM, i.e., dipolar DM and sterile
neutrino DM, whose coupling to the photon would induce a rotation of the
polarization plane of the CMB, and try to study their properties through CMB
birefringence.

###  IV.1 Dipolar dark matter

The effective Lagrangian for the coupling of the electromagnetic field
$F^{\mu\nu}$ with a Dirac fermion that possesses a magnetic dipole moment
$\mathcal{M}$ and an electric dipole moment $\mathcal{D}$ is as follows:

$\mathcal{L}_{\rm
DDM}=-\frac{i}{2}\bar{\psi}\sigma_{\rm\mu\nu}(\mathcal{M}+\gamma^{5}\mathcal{D})\psi
F^{\mu\nu},$ (18)

where $\sigma^{\mu\nu}$ is the commutator of two Dirac matrices,
$\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$. Regarding Majorana
fermions, only non-zero transition multipole moments between different mass
eigenstates can be defined, and their interaction with photons is described by
Masso:2009mu ; Lee:2014koa

$\mathcal{L}_{\rm
DDM}=-\frac{i}{2}\bar{\psi}_{2}\sigma_{\rm\mu\nu}(\mathcal{M}_{12}+\gamma^{5}\mathcal{D}_{12})\psi_{1}F^{\mu\nu}+H.c.,$
(19)

where $\mathcal{M}_{12}$ is a transition magnetic moment, and
$\mathcal{D}_{12}$ is a transition electric moment. This possible interaction
between photon and dipolar DM opens up a new window to explore the features of
this candidate of DM. Here, we focus on the Majorana DM which interacts with
CMB photons via transition magnetic dipole moment and study its properties
using the CB effect of the CMB. To this end, we consider two singlet Majorana
fermions $\chi_{1}$ and $\chi_{2}$, with mass splitting $\delta$, which couple
to photons by the magnetic dipole interaction Lagrangian Davidson:2005cs

$\mathcal{L}_{\text{\tiny{DDM}}}=-\frac{i}{2}\,\mathcal{M}_{12}\,\bar{\chi}_{1}\sigma_{\rm\mu\nu}\chi_{2}F^{\mu\nu}+H.C..$
(20)

Assuming the singlet Majorana fermions to be the right-handed neutrinos
$(\chi\,=\,\psi_{R}+\psi_{R}^{c})$, the Lagrangian (20) casts into

$\mathcal{L}_{\text{\tiny{DDM}}}=-\frac{i}{2}\,\,\mathcal{M}_{12}\,(\,\bar{\psi^{c}}_{1}\sigma_{\rm\mu\nu}P_{R}\psi_{2}\,\,F^{\mu\nu}\,\,+\,\,\bar{\psi}_{1}\sigma_{\rm\mu\nu}P_{L}\psi^{c}_{2}\,\,F^{\mu\nu})+H.C.,$
(21)

where $P_{R}=\frac{1}{2}(1+\gamma^{5})$, $P_{L}=\frac{1}{2}(1-\gamma^{5})$ and
$\psi^{c}=-i\gamma_{2}\psi^{\star}$.

The Feynman diagram corresponding to this interaction at the lowest order is
shown in Fig.1.

Figure 1: The typical diagrams for photon-DM scattering

To investigate the possible effects of this interaction on the CMB
polarization at the forward scattering level, one needs to calculate
$\langle[H_{I}^{0}(0),D_{ij}({\bf{k}})]\rangle$ which is obtained as
Mahmoudi:2018lll

$i\langle[H_{I}^{0}(0),D_{ij}({\bf{k}})]\rangle=i\int
d{\bf{q}}\,n_{\text{\tiny{DM}}}({\bf{x}},{\bf{q}})(\delta_{is}\rho_{s^{\prime}j}({\bf{k}})-\delta_{js^{\prime}}\rho_{is}({\bf{k}}))(2\pi)^{3}\delta^{(3)}(0){M}\,\,|_{{\tiny{q^{\prime}=q,\,\,p^{\prime}=p=k,r=r^{\prime}}}},$
(22)

where
${d\bf{q}}\equiv\frac{d^{3}\bf{q}}{(2\pi)^{3}}\frac{m_{\text{{\tiny{DM}}}}}{q^{0}}$,
$\rho_{ss^{\prime}}({\bf{k}})$ indicates the elements of the CMB photon
density matrix and $n_{\text{\tiny{DM}}}({\bf{x}},{\bf{q}})$ is the number
density of DM. Moreover, ${M}$ denotes the total Feynman amplitude given by
the following equation in the non-relativistic limit:

$\displaystyle{M}$ $\displaystyle=$
$\displaystyle-\mathcal{M}_{12}^{2}\frac{(2k.q)^{2}}{(2k.q)^{2}-(m_{\text{\tiny{DM}}_{2}}^{2}-m_{\text{\tiny{DM}}_{1}}^{2})^{2}}\Big{(}{\bf{k}}\cdot(\vec{\epsilon}_{s^{\prime}}\times\vec{\epsilon}_{s})-k^{0}v(\vec{\epsilon}_{s^{\prime}}\times\vec{\epsilon}_{s})\cdot{\hat{{\bf{v}}}}\Big{)}+(1\leftrightarrow
2),$ (23)

where $v=|\vec{q}|/m_{\text{\tiny{DM}}}$ denotes the velocity of DM and
$\vec{\epsilon}_{s}(k)$’s are the photon polarization vectors with $s=1,2$ for
two physical transverse polarization of a free photon. Considering the case
that $\delta=m_{\text{\tiny{DM}}_{2}}-m_{\text{\tiny{DM}}_{1}}\ll k^{0}$, (23)
can be estimated as follows

${M}\simeq-\mathcal{M}_{12}^{2}\,\,{\bf{k}}\cdot(\vec{\epsilon}_{s^{\prime}}\times\vec{\epsilon}_{s})\,+\,(1\,\leftrightarrow\,2),$
(24)

where since the order of the second term is smaller than the first one due to
the presence of $v$, we ignore the terms proportional to the DM velocity.
Moreover, it is important to note that in the case $\delta\gg k^{0}$, the
contribution of the photon-dipolar DM scattering on the CMB polarization will
be suppressed as $(\frac{k^{0}}{\delta})^{2}$ compared to the case $\delta\ll
k^{0}$ and therefore, we will not consider this case for the rest of the paper
(see appendix A for more details).

By substituting (24) in (22) and using (2), the time evolution of the density
matrix element can be written as

$\frac{d\rho_{ij}}{dt}=-i\mathcal{M}^{2}\,\,\big{(}n_{\text{\tiny{DM}}_{1}}({\bf{x}})+n_{\text{\tiny{DM}}_{2}}({\bf{x}})\big{)}(\delta_{is}\rho_{s^{\prime}j}({\bf{k}})-\delta_{js^{\prime}}\rho_{is}({\bf{k}}))(\vec{\epsilon}_{s^{\prime}}\times\vec{\epsilon}_{s})\cdot\hat{{\bf{k}}},$
(25)

where $\mathcal{M}_{12}^{2}=\mathcal{M}_{21}^{2}=\mathcal{M}^{2}$,
$\hat{{\bf{k}}}={\bf{k}}/k^{0}$ and the DM number density
$n_{\text{\tiny{DM}}_{\text{\tiny{i}}}}$ ($\text{i}=1,2$) is

$n_{\text{\tiny{DM}}_{\text{\tiny{i}}}}({\bf{x}})=\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\,\,n_{\text{\tiny{DM}}_{\text{\tiny{i}}}}({\bf{x}},{\bf{q}}).$
(26)

Since
$n_{\text{\tiny{DM}}_{1}}({\bf{x}})+n_{\text{\tiny{DM}}_{2}}({\bf{x}})=n_{\text{\tiny{DM}}}({\bf{x}})$,
(25) is reduced to

$\frac{d\rho_{ij}}{dt}=-i\mathcal{M}^{2}\,\,n_{\text{\tiny{DM}}}({\bf{x}})\,\,(\delta_{is}\rho_{s^{\prime}j}({\bf{k}})-\delta_{js^{\prime}}\rho_{is}({\bf{k}}))(\vec{\epsilon}_{s^{\prime}}\times\vec{\epsilon}_{s})\cdot\hat{{\bf{k}}},$
(27)

and, consequently, the Stokes parameters evolve as

$\displaystyle\frac{dI}{dt}$ $\displaystyle=$ $\displaystyle C_{e\gamma}^{I},$
(28) $\displaystyle\frac{d}{dt}(Q\pm iU)$ $\displaystyle=$ $\displaystyle
C_{e\gamma}^{\pm}\mp i\dot{\tau}_{\text{\tiny{DM}}}(Q\pm iU),$ (29)
$\displaystyle\frac{dV}{dt}$ $\displaystyle=$ $\displaystyle C_{e\gamma}^{V},$
(30)

where $C_{e\gamma}^{I}$,$C_{e\gamma}^{V}$ and $C_{e\gamma}^{\pm}$ show the
contribution of Thomson scattering Kosowsky:1994cy  and
$\dot{\tau}_{\text{DM}}$ is defined as follows:

$\dot{\tau}_{\text{\tiny{DM}}}=\frac{\mathcal{M}^{2}}{m_{\text{\tiny{DM}}}}\,\rho_{\text{\tiny{DM}}},$
(31)

in which $\rho_{\text{\tiny{DM}}}$ is the DM mass density. The second term in
the right-hand side of (29) shows that the photon-dipolar DM forward
scattering affects the time evolution of the linear polarization of the CMB

$\dfrac{d}{d\eta}\Delta^{\pm\text{S}}_{\text{P}}+iK\mu\Delta^{\pm\text{S}}_{\text{P}}=\dot{\tau_{\text{e}}}\left[-\Delta^{\pm\text{S}}_{\text{P}}-\dfrac{1}{2}[1-P_{2}(\mu)]\Pi\right]\mp
ia(\eta)\dot{\tau}_{\text{\tiny{DM}}}\Delta^{\pm\text{S}}_{\text{P}},$ (32)

which leads to the following equation of polarization anisotropy:

$\dfrac{d}{d\eta}\left[\Delta^{\pm\text{S}}_{\text{P}}e^{iK\mu\eta\pm
i\tilde{\tau}_{\text{\tiny{DM}}}+\tilde{\tau}_{\text{e}}}\right]=-\dfrac{1}{2}e^{iK\mu\eta\pm
i\tilde{\tau}_{\text{\tiny{DM}}}+\tilde{\tau}_{\text{e}}}\dot{\tau_{\text{e}}}[1-P_{2}(\mu)]\Pi,$
(33)

where

$\tilde{\tau}_{\text{\tiny{DM}}}(\eta,\mu)\equiv\int_{0}^{\eta}d\eta
a(\eta)\dot{\tau}_{\text{\tiny{DM}}},\,\,\,\,\,\,\,\,\,\,\,\,\tilde{\tau}_{\text{e}}(\eta)\equiv\int_{0}^{\eta}d\eta\,a(\eta)\,\dot{\tau}_{\text{e}}.$
(34)

After calculation of $\Delta^{\pm\text{S}}_{P}$ and using (9) and (10), the
E-mode and B-mode polarizations produced through the dipolar DM-photon
interaction will be obtained as follows:

$\displaystyle\Delta^{\text{S}}_{\text{E}}(\eta_{0},K,\mu)$ $\displaystyle=$
$\displaystyle-\dfrac{3}{4}\int_{0}^{\eta_{0}}d\eta\,g_{\text{e}}(\eta)\Pi(\eta,K)\partial^{2}_{\mu}[(1-\mu^{2})e^{ix\mu}\cos\tau_{\text{\tiny{DM}}}],$
(35) $\displaystyle\Delta^{\text{S}}_{\text{B}}(\eta_{0},K,\mu)$
$\displaystyle=$
$\displaystyle\dfrac{3}{4}\int_{0}^{\eta_{0}}d\eta\,g_{\text{e}}(\eta)\Pi(\eta,K)\partial^{2}_{\mu}[(1-\mu^{2})e^{ix\mu}\sin\tau_{\text{\tiny{DM}}}],$
(36)

where $x=K(\eta_{0}-\eta)$ and
$g_{\text{e}}(\eta)\equiv\dot{\tau}_{\text{e}}e^{-\tau_{\text{e}}}$ is the
visibility function of the electron and describes the probability that a
photon scattered at epoch $\eta$ reaches the observer at the present time,
$\eta_{0}$ Dodelson:2003ft . Finally, the power spectrum of the E and B-modes
will be obtained by integrating over the initial power spectrum of the metric
perturbation as:

$\displaystyle C_{EE,\ell}^{(\rm S)}\Big{|}_{\rm DM}$ $\displaystyle=$
$\displaystyle(4\pi)^{2}\frac{(\ell+2)!}{(\ell-2)!}\int
d^{3}KP_{S}(K)\left[\dfrac{3}{4}\int_{0}^{\eta_{0}}d\eta\,g_{e}(\eta)\,\Pi(K,\eta)\,\dfrac{j_{\ell}(x)}{x^{2}}\cos(\tau_{DM})\right]^{2},$
(37) $\displaystyle C_{BB,\ell}^{(\rm S)}\Big{|}_{\rm DM}$ $\displaystyle=$
$\displaystyle(4\pi)^{2}\frac{(\ell+2)!}{(\ell-2)!}\int
d^{3}KP_{S}(K)\left[\dfrac{3}{4}\int_{0}^{\eta_{0}}d\eta\,g_{e}(\eta)\,\Pi(K,\eta)\,\dfrac{j_{\ell}(x)}{x^{2}}\sin(\tau_{DM})\right]^{2}.$
(38)

Moreover, the cross-power spectrum will be as follows:

$\displaystyle C_{EB,\ell}^{(\rm S)}\Big{|}_{\rm DM}$ $\displaystyle=$
$\displaystyle\dfrac{(4\pi)^{2}}{4}\frac{(\ell+2)!}{(\ell-2)!}\int
d^{3}KP_{S}(K)\left[\dfrac{3}{4}\int_{0}^{\eta_{0}}d\eta\,g_{e}(\eta)\,\Pi(K,\eta)\,\dfrac{j_{\ell}(x)}{x^{2}}\sin(2\tau_{DM})\right]^{2}.$
(39)

Since the standard E-mode coming from the Compton scattering in the CMB, i.e.,
$\bar{C}_{EE,\ell}^{(\rm S)}$, is

$\displaystyle\bar{C}_{EE,\ell}^{(\rm S)}\Big{|}_{\rm DM}$ $\displaystyle=$
$\displaystyle(4\pi)^{2}\frac{(\ell+2)!}{(\ell-2)!}\int
d^{3}KP_{S}(K)\left[\dfrac{3}{4}\int_{0}^{\eta_{0}}d\eta\,g_{e}(\eta)\,\Pi(K,\eta)\,\dfrac{j_{\ell}(x)}{x^{2}}\right]^{2},$
(40)

Eqs. (37), (38)and (39) can be approximated as follows:

$\displaystyle C_{EE,\ell}^{(\rm S)}\Big{|}_{\rm DM}$ $\displaystyle\cong$
$\displaystyle\cos^{2}(\tau_{DM})\bar{C}_{EE,\ell}^{(\rm S)},$ $\displaystyle
C_{BB,\ell}^{(\rm S)}\Big{|}_{\rm DM}$ $\displaystyle\cong$
$\displaystyle\sin^{2}(\tau_{DM})\bar{C}_{EE,\ell}^{(\rm S)},$ $\displaystyle
C_{EB,\ell}^{(\rm S)}\Big{|}_{\rm DM}$ $\displaystyle\cong$
$\displaystyle\dfrac{1}{4}\sin^{2}(2\tau_{DM})\bar{C}_{EB,\ell}^{(\rm S)}.$
(41)

The term $\tau_{DM}$ is the effective opacity of DM produced by the DM-photon
interaction from the last scattering surface to the present time and is
determined by (31) and (34). It is more convenient to express Eq.(34) in terms
of the redshift $z$ which will be as follows:

$\tau_{\text{\tiny{DM}}}(z)=\,\frac{{\mathcal{M}}^{2}}{m_{\text{\tiny{DM}}}}\,\rho_{\text{\tiny{DM}}}^{0}\,\int_{0}^{z}dz^{{}^{\prime}}\,\dfrac{(1+z^{{}^{\prime}})^{2}}{H(z^{{}^{\prime}})}.$
(42)

To arrive the above equation, we used

$\rho_{\text{\tiny{DM}}}=\rho_{\text{\tiny{DM}}}^{0}(1+z)^{3},$ (43)

where $\rho_{\text{\tiny{DM}}}^{0}$ is the mass density of DM in the present
time and

$a\,d\eta=-\frac{dz}{H(z)(1+z)},$ (44)

where $H(z)$ is the Hubble parameter. Making use of the Friedmann equation in
the matter dominated era

$\frac{H^{2}}{H_{0}^{2}}=\Omega_{M}^{0}(1+z)^{3}+\Omega_{\Lambda}^{0},$ (45)

Figure 2: (color online). Dipolar DM parameter space $[m,\mathcal{M}]$. The
shaded blue region is the place where the viable Dirac candidates must lie in,
below the solid lines and outside the long-dashed lines. The short-dashed
relic abundance curve, which is obtained by considering the standard freeze-
out, indicates where the DM would meet a cosmological density $\Omega
h^{2}=0.135$, assuming no particle-antiparticle asymmetry, and no interactions
with standard-model particles except the dipole coupling to photons. Note that
the EGRET and GLAST curves constrain the combination
$(\mathcal{D}^{4}+\mathcal{M}^{4})^{1/4}$; the perturbative and unitarity
curves apply to the stronger of $(\mathcal{D},\mathcal{M})$; while all other
curves restrict $(\mathcal{D}^{2}+\mathcal{M}^{2})^{1/2}$. (see
Sigurdson:2004zp  for more detail). The plot of cosmic birefringence
constrains $\mathcal{M}$. The shaded pink region is the area excluded from the
dipolar DM parameter space by the results of CB for the Majorana dipolar DM.
Moreover, the pale pink area marked by the dashed-dotted line represents the
uncertainty in the upper limit obtained due to this effect.

The effective opacity (42) becomes

$\tau_{DM}=\frac{{\mathcal{M}}^{2}}{m_{\text{\tiny{DM}}}}\,\rho_{\text{\tiny{DM}}}^{0}\frac{2H(z^{\prime})}{3\Omega_{M}^{0}H_{0}^{2}}\Big{|}^{z^{\prime}=z}_{z^{\prime}=0},$
(46)

where $H_{0}\,=\,(67.4\,\pm\,0.5)\,\text{km}\text{s}^{-1}\text{Mpc}^{-1}$,
$\Omega_{M}^{0}\,=\,0.315\,\pm\,0.007$, $\Omega_{\Lambda}^{0}\approx 0.69$
Planck:2018vyg . The redshift dependence of the above equation indicates that
the maximum value of $\tau_{DM}$ occurs near the last scattering surface and
its approximate value is as follows:

$\tilde{\tau}_{DM}\approx 4.6\times
10^{25}\,\left(\frac{\mathcal{M}}{e\,\text{cm}}\right)^{2}\,\left(\frac{GeV}{m_{\text{\tiny{DM}}}}\right)\,\,\,\,\,\,\,\,\,,\bigg{(}\frac{\rho_{\text{\tiny{DM}}}^{0}}{2.5\times
10^{-30\,\,g/cm^{3}}}\bigg{)}\,\,\bigg{(}\frac{z^{\prime}}{10^{3}}\bigg{)}.$
(47)

Now, using the equation between the CB angle $\beta$ and the maximum value of
the effective opacity Mohammadi:2021xoh

$\beta\,\approx\,\frac{1}{2}\,\tilde{\tau}_{DM},$ (48)

we can estimate the contribution of the dipolar DM interaction with the photon
in producing the CB effect. For instance, the CB angle of the CMB due to the
interaction with dipolar DM whose mass is around $1\text{MeV}$ will be
approximated as

$\beta\big{|}_{\text{\tiny}{DM}}\approx 2.0\times
10^{-3}\,rad\,\,\,\,\,\left(\frac{\mathcal{M}}{3\times
10^{-16}\,e\,cm}\right)^{2}\,\,\,\left(\frac{10^{-3}GeV}{m_{\text{\tiny{DM}}}}\right).$
(49)

Considering the value of the CB angle reported by using the Planck data,
$\beta=0.30^{\circ}\pm 0.11^{\circ}$ ($68\%\text{C.L.}$) Diego-
Palazuelos:2022dsq , we realize that dipolar DM with the mentioned properties
can approximately compensate for about $(40\pm 13)\%$ of the CB angle of the
CMB, where the uncertainty originates from the uncertainty on $\beta$. It is
important to note that the chosen values for mass and magnetic moment in Eq.
(49) are threshold values, which means that according to cosmological
constraints, the mass of dipolar DM must be larger than $1MeV$ Burles:1999zt ,
and also based on the reported results concerning the magnetic moment of DM,
the value of this quantity is approximately less than $10^{-15}ecm$, (see Fig.
2) . Therefore, using relations Eqs. (47) and (48), one can easily find that
with the increase in mass and decrease in coupling constant, the contribution
of this type of DM on the generation of the CB effect will be less, and hence,
it can contribute to producing CB up to the mentioned value at most.

Another perspective that we can follow concerning this issue is putting a
constraint on the magnetic dipole moment depending on the mass of DM
particles. In this regard, by combining Eqs. (47) and (48),

$\mathcal{M}\approx\big{(}\frac{1}{2.3\times
10^{25}}\big{)}^{1/2}\,\sqrt{\beta}\,\,\,e\text{\,cm}\,\,\,\,\,\,\,\,\,\,\,\,\big{(}\frac{m_{\text{\tiny{DM}}}}{1\,GeV}\big{)}^{1/2},$
(50)

and substituting the value of CB angle reported by using the Planck data
release, $\beta=0.30^{\circ}\pm 0.11^{\circ}$ ($68\%\text{C.L.}$), the
following constraint will be placed on the phase parameters of the dipolar DM

$\mathcal{M}\leqslant(1.4\pm 0.23)\times
10^{-14}\,e\text{\,cm}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\big{(}\frac{m_{\text{\tiny{DM}}}}{1\,GeV}\big{)}^{1/2},$
(51)

where the uncertainty on $\mathcal{M}$ originates from the uncertainty on
$\beta$, i.e.

$\Delta\mathcal{M}\approx\big{(}\frac{\beta}{2.3\times
10^{25}}\big{)}^{1/2}\,\,\frac{\Delta\beta}{2\beta}\,\,\,\,\,\,\,\,e\text{\,cm}\,\,\,\,\big{(}\frac{m_{\text{\tiny{DM}}}}{1\,GeV}\big{)}^{1/2}$
(52)

For instance, our results put a bound on the magnetic dipole moment about
$\mathcal{M}\leqslant(7\pm 1.1)\times 10^{-16}\,e\text{\,cm}$ for dipolar DM
particles whose mass is around $m_{\text{\tiny{DM}}}\approx 3\,MeV$. The full
mass dependence of this result is shown in Fig.2. This figure is originally
based on the constraints on the dipolar DM parameter space that come from some
theoretical and experimental research and are adapted from Sigurdson:2004zp .
The Viable Dirac dipolar DM must lie in the shaded blue region, below the
solid lines, and outside the long-dashed lines (see Sigurdson:2004zp  for more
detail). The shaded pink region is the region excluded from the dipolar DM
parameter space by the CB results. It is important to mention that the result
obtained from considering the CB effect is related to Majorana dipolar DM
particles, while other results are dedicated to Dirac dipolar DM. Indeed, we
put a constraint on the sub-GeV Majorana dipolar DM through the CB effect of
the CMB photons. It is also notable that the constraints obtained on Majorana
dipolar DM, including the constraints that come from the direct detection
experiments Masso:2009mu , study DM particles whose mass is around GeV or
higher, while the constraint that we obtain here is regarding the sub-GeV DM
particles. Before ending this section, it is worth mentioning that although an
experiment like W boson mass excludes the sub-GeV Dirac dipolar DM with a
magnetic moment larger than $7\times 10^{-16}\,e\,\text{cm}$, part of this
region will be accessible for Majorana dipolar DM based on the CB effect of
the CMB.

###  IV.2 Sterile Neutrino dark matter

The existence of the right-handed sterile neutrinos is elegantly formulated in
the seesaw model. In the framework of type-I seesaw model, the SM is extended
by at least two heavy sterile neutrino singlets $\nu_{\text{\tiny{R}}}^{i}$
($i$ indicates the generation) which mix with the SM neutrinos through a
mixing angle $\theta$ and form the neutrino physical states as follows:

$\displaystyle
N=V^{\dagger}_{\text{\tiny{N}}}\nu_{\text{\tiny{R}}}+U^{\dagger}_{N}\theta\nu^{c}_{\text{\tiny{L}}}+h.c.\,,~{}~{}~{}~{}\text{and}~{}~{}~{}~{}\upnu=V^{\dagger}_{\nu}\nu_{\text{\tiny{L}}}-U^{\dagger}_{\nu}\theta\nu_{\text{\tiny{R}}}^{c}+h.c.\,,$
(53)

where $V$, $\theta$ and $U$ contain information about the mixing angle
Haghighat:2019rht . Indeed, $V_{\nu}$ is the usual neutrino mixing matrix
connecting the observed light mass eigenstates to the active flavor
eigenstates:

$V_{\nu}\equiv(1-\frac{1}{2}\theta\theta^{\dagger})U_{\nu},$ (54)

and $U_{\nu}$ is the unitary part of neutrino mixing matrix Boyarsky:2018tvu .
Meanwhile, the corresponding parameters in the sterile sector are
$V_{\text{\tiny{N}}}$ and $U_{\text{\tiny{N}}}$ and the active-sterile mixing
angle is

$\Theta\equiv\theta U_{\text{\tiny{N}}}^{\star}.$ (55)

In the SM, neutrinos interact with other particles only via the weak
interaction,

$-\frac{g}{\sqrt{2}}\overline{{\nu_{l}}_{\tiny{L}}}\gamma^{\mu}l_{L}W^{+}_{\mu}-\frac{g}{\sqrt{2}}\overline{l_{L}}\gamma^{\mu}{\nu_{l}}_{L}W^{-}_{\mu}-\frac{g}{2\cos\theta_{W}}\overline{{\nu_{l}}_{L}}\gamma^{\mu}{\nu_{l}}_{L}Z_{\mu},$
(56)

where $g$ is the gauge coupling constant, $\theta_{W}$ stands for the Weinberg
angle, $l=e,\mu,\tau$ and, ${\nu_{l}}_{L}$ denotes the flavor state of left-
handed SM neutrinos. Using Eqs.(53), one can express the neutrino flavor
eigenstates ($\nu_{L}$) in terms of neutrino physical states, i.e.
$\nu_{L}=V_{\nu}\upnu+\Theta N$ and therefore, in the most general form, the
mass eigenstate sterile neutrinos($N$) can interact with the SM particles
through the mixing angle as follows Boyarsky:2018tvu :

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\,\,\sum_{l}-\frac{g}{\sqrt{2}}\bar{N}\,\Theta^{\dagger}\,\gamma^{\mu}\,l_{\text{\tiny{L}}}\,W_{\mu}^{+}\,-\sum_{l}\frac{g}{\sqrt{2}}\,\bar{l}_{\text{\tiny{L}}}\,\gamma^{\mu}\,\Theta\,N\,W_{\mu}^{-}-\frac{g}{2\cos\theta_{\text{\tiny{W}}}}\,\bar{N}\,\Theta^{\dagger}\,\gamma^{\mu}\,\nu_{l_{\text{\tiny{L}}}}\,Z_{\mu}\,$
(57)
$\displaystyle-\frac{g}{2\cos\theta_{\text{\tiny{W}}}}\,\bar{\nu}_{l_{\text{\tiny{L}}}}\,\gamma^{\mu}\,\Theta\,N\,Z_{\mu}-\frac{g}{\sqrt{2}}\,\frac{M_{\text{\tiny{N}}}}{m_{\text{\tiny{W}}}}\,\Theta\,h\,\bar{\nu}_{l_{\text{\tiny{L}}}}\,N\,-\frac{g}{\sqrt{2}}\,\frac{M_{\text{\tiny{N}}}}{m_{\text{\tiny{W}}}}\,\Theta^{\dagger}\,h\,\bar{N}\,\nu_{l_{\text{\tiny{L}}}}.$

where $h$ is physical Higgs field, $M_{N}$ denotes the mass of sterile
neutrino and $m_{W}$ stands for the mass of $W$ boson. Note that depending on
models and the sterile neutrino production mechanism, and considering the
astrophysical constraints, one can find different bounds on the mixing angle.
For instance, some considerations which predict active generation of such
particles in the early Universe constrain $\theta^{2}\ll 10^{-8}$ from the
total DM relic density and the absence of X-ray signal from sterile neutrino
decayBoyarsky:2018tvu ; Dolgov:2002wy . However, regarding some models with a
hidden sector coupled to the sterile neutrino, these bounds can be extended to
$\theta^{2}\leqslant 10^{-1}$ from the total DM relic density Bezrukov:2017ike
.

Based on some reports regarding the galaxy phase space density, universal
galaxy surface density and the DM density, the mass eigenstate sterile
neutrinos (N) can be fit to a warm DM scenario Dodelson:1993je . Here, we
study the CB of the CMB due to its interaction with sterile neutrino DM. In
the context of the seesaw model, photons can scatter from sterile neutrinos at
a one-loop level with a lepton (or antilepton) and weak gauge bosons
propagating in the loop. As the authors have shown in Haghighat:2019rht , the
sterile neutrino-CMB interaction given by Fig.3 can affect the CMB
polarization features and modify the power spectrum of the B- mode
polarization. Indeed, to examine the effects of the photon-sterile neutrino
interaction on the polarization of the CMB photons, we take (57) and (2) into
account to find the time evolution of the density matrix components as follows
(see appendix B for more detail) Haghighat:2019rht :

$\displaystyle\frac{d}{dt}\rho_{ij}(k)\,$ $\displaystyle=$
$\displaystyle\,-\frac{\sqrt{2}}{12\,\pi\,k^{0}}\alpha\,\theta^{2}\,{G}_{\text{\tiny{F}}}\int
d{\bf{q}}\,\,(\delta_{is}\rho_{s^{\prime}j}(k)-\delta_{js^{\prime}}\rho_{is}(k))\,f_{\text{\tiny{DM}}}({\bf{x}},{\bf{q}})\,\bar{u}_{r}(q)\,\,(1-\gamma^{5})$
(58)
$\displaystyle(q\cdot\epsilon_{s}\,\,\not{\epsilon}_{s^{{}^{\prime}}}\,+\,q\cdot\epsilon_{s^{{}^{\prime}}}\,\,\not{\epsilon}_{s})\,u_{r}(q)+\frac{\sqrt{2}}{24\,\pi\,k^{0}}\alpha\,\theta^{2}\,{G}_{\text{\tiny{F}}}\int
d{\bf{q}}\,\,(\delta_{is}\rho_{s^{\prime}j}(k)-\delta_{js^{\prime}}\rho_{is}(k))\,$
$\displaystyle
f_{\text{\tiny{DM}}}({\bf{x}},{\bf{q}})\,\bar{u}_{r}(q)\,(1-\gamma^{5})\,\not{k}\,(\not{\epsilon}_{s^{{}^{\prime}}}\,\not{\epsilon}_{s}\,-\,\not{\epsilon}_{s}\,\not{\epsilon_{s^{{}^{\prime}}}})\,u_{r}(q).$

where $\epsilon_{s}(k)$ with $s=1,2$ are the photon polarization 4-vectors of
two physical transverse polarizations, while $u_{r}(q)$ and $v_{r}(q)$ are the
Dirac spinors. Furthermore, $f_{\text{\tiny{DM}}}$ denotes the distribution
function of DM, and $G_{F}$ and $\alpha$ are the Fermi coupling constant and
electromagnetic fine structure constant, respectively. One can reconstruct the
Stokes parameters through the density matrix elements and using the following
identities

$\displaystyle\sigma_{\mu\nu}\gamma_{\alpha}$ $\displaystyle=$
$\displaystyle-i(\delta_{\mu\alpha}\gamma_{\nu}+\delta_{\nu\alpha}\gamma_{\mu}+\epsilon_{\mu\nu\alpha\lambda}\gamma^{\lambda}\gamma^{5}),$
$\displaystyle\bar{u}_{r}(q)\gamma^{\mu}\,u_{r}(q)$ $\displaystyle=$
$\displaystyle 2\frac{q^{\mu}}{m_{DM}},$
$\displaystyle\bar{u}_{r}(q)\gamma^{\mu}(1\pm\gamma^{5})\,u_{r}(q)$
$\displaystyle=$ $\displaystyle 2\frac{q^{\mu}}{m_{DM}},$ (59)

where the completely antisymmetric alternating symbol
$\epsilon_{\mu\nu\alpha\lambda}$ is equal to $+1$ for
($\mu,\nu,\alpha,\lambda$) an even permutation of $(0,1,2,3)$, is equal to
$-1$ for an odd permutation, and vanishes if two or more indices are the same.
Consequently, reconstruction of the Stokes parameters shows that this
interaction can affect the evolution of the linear polarization of the CMB as
follows:

$\frac{d}{d\eta}\,\Delta_{P}^{\pm(S)}\,+\,i\,K\,\mu\,\Delta_{P}^{\pm(S)}\,=\,C_{e\gamma}^{\pm}\,\mp\,i\,a(\eta)\dot{\tau}_{\text{\tiny{DM}}}\,\Delta_{P}^{\pm},$
(60)

in which $\dot{\tau}_{\text{\tiny{DM}}}$ considered for the contribution of
the photon-sterile neutrino scattering can be obtained as

$\displaystyle\dot{\tau}_{\text{\tiny{DM}}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{3\pi
k^{0}\,m_{\text{\tiny{DM}}}}\,\,\alpha\,\,{G}_{\text{\tiny{F}}}\,\theta^{2}\,\int\,d{\bf{q}}\,f_{\text{\tiny{DM}}}({\bf
x},{\bf
q})\,\times(\varepsilon_{\mu\,\nu\,\rho\,\sigma}\epsilon_{2}^{\mu}\,\epsilon_{1}^{\nu}\,k^{\rho}\,q^{\sigma}),$
(61)

where it can be reduced to

$\displaystyle\dot{\tau}_{\text{\tiny{DM}}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{3\pi
k^{0}\,m_{\text{\tiny{DM}}}}\,\,\alpha\,\,{G}_{\text{\tiny{F}}}\,\theta^{2}\int
d\mathbf{q}\,f_{\text{\tiny{DM}}}({\bf x},{\bf q})\times\left[q^{0}{\bf
k}\cdot(\epsilon_{1}\times\epsilon_{2})+k^{0}{\bf
q}\cdot(\epsilon_{1}\times\epsilon_{2})\right]$ (62) $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{3\pi}\,\,\alpha\,\,{G}_{\text{\tiny{F}}}\,\theta^{2}\,n_{\text{\tiny{DM}}}\left[1+\langle{\bf
v}\rangle\cdot(\epsilon_{1}\times\epsilon_{2})\right]\approx\frac{\sqrt{2}}{3\pi}\,\,\alpha\,\,{G}_{\text{\tiny{F}}}\,\theta^{2}\,n_{\text{\tiny{DM}}},$

where ${\bf k}\cdot(\epsilon_{1}\times\epsilon_{2})=|{\bf k}|$, the DM number
density
$n_{\text{\tiny{DM}}}=\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}f_{\text{\tiny{DM}}}({\bf
x},{\bf q})$, and $\langle{\bf v}\rangle$ is the average velocity of DM
particles. Since the average velocity of DM is small, the dominated
contribution of this scattering to photon polarization comes from the first
term and thus we ignore the term including $\langle{\bf v}\rangle$.

Figure 3: The representative Feynman diagrams represent the photon-Sterile
neutrino scattering, where $l=e,\mu,\tau$ and $\bar{l}$ indicates the anti-
particle of $l$.

Using (34), (44) and (45), we arrive at the following equation for the
effective opacity $\tau_{\text{\tiny{DM}}}$

$\tau_{\text{\tiny{DM}}}=\frac{\sqrt{2}}{3\pi\,m_{\text{\tiny{DM}}}}\,\,\alpha\,\,{G}_{\text{\tiny{F}}}\,\theta^{2}\,\rho_{\text{\tiny{DM}}}^{0}\,\frac{2H(z^{\prime})}{3\Omega_{M}^{0}H_{0}^{2}}\Big{|}^{z^{\prime}=z}_{z^{\prime}=0},$
(63)

where we have used the fact that
$\rho_{\text{\tiny{DM}}}\,=\,\rho_{\text{\tiny{DM}}}^{0}(1+z)^{3}$ in which
$\rho_{\text{\tiny{DM}}}^{0}$ is the mass density of DM in the present time.
Now, we try to make an estimate of the maximum value $\tau_{\text{\tiny{DM}}}$
near the last scattering which leads to the following equation:

$\tilde{\tau}_{\text{\tiny{DM}}}\approx 3\times
10^{-9}\,\theta^{2}\,\left(\frac{GeV}{m_{\text{\tiny{DM}}}}\right)\,\bigg{(}\frac{\rho_{\text{\tiny{DM}}}^{0}}{10^{-47}\,GeV^{4}}\bigg{)}\,\,\bigg{(}\frac{z^{\prime}}{10^{3}}\bigg{)}.$
(64)

Hence, the CB angle of the CMB due to the interaction with the sterile
neutrinos can be approximated as

$\beta\approx 1.5\times
10^{-9}\,\theta^{2}\,\left(\frac{GeV}{m_{\text{\tiny{DM}}}}\right).$ (65)

Before to proceed, it is worth discussing the approximate CB angle which can
be caused by this sort of interaction. Since according to some cosmological
constraints, the mass of sterile neutrinos must be larger than $100eV$
Boyarsky:2018tvu  and based on the reported results regarding the mixing
angle, the value of this quantity is approximately less than $10^{-3}$, we
come to the conclusion that by considering the value of the CB angle,
$\beta=0.30^{\circ}\pm 0.11^{\circ}$, around $(0.30\pm 0.10)\%$ of the CB
angle can be caused by the interaction with the sterile neutrino DM (if it
exists).

Figure 4: (color online). The constraints on the sterile neutrino DM parameter
space assuming a standard cosmology below the temperature when neutrino
oscillations occur are adapted from Buckley:2013bha . The shaded brown region
is the area excluded by the CB effect of the CMB.

Moreover, using the mentioned value reported for the CB by the Planck
collaboration, one can put a constraint on the parameter space of the sterile
neutrino DM:

$\theta^{2}\leq\,(3.3\pm
1.1)\,(rad)^{2}\,\,\,\,\,\,(\frac{m_{\text{\tiny{DM}}}}{1\,\text{KeV}})\,$
(66)

The parameter space of the sterile neutrino DM $[m,\sin^{2}\theta]$ is
depicted in Fig. 4. This figure, adapted from Buckley:2013bha , is originally
based on the summary of astrophysical constraints on the parameter space,
$m_{st}-\theta$ plane, for the sterile neutrino DM. As this figure shows, the
constraint on the sterile neutrino parameter space due to the CB effect is
placed in the region that has already been excluded by the X-ray experiment.
However, we should point out that the values of mass ($m_{\text{\tiny{DM}}}$)
and the mixing angle ($\theta$) could depend on cosmological production
scenarios. In fact, one of the most important issues of sterile neutrinos is
to address the question of how they have been produced in the early universe.
Generally, sterile neutrinos can be produced by neutrino oscillation in the
primordial plasma via a tiny active-sterile neutrino mixing angle $\theta$, as
first described by Dodelson and Widrow (DW) Dodelson:1993je . Various current
astrophysical observations impose severe constraints on the sterile neutrino
DM which is generated by the DW mechanism Boyarsky:2005us ; Boyarsky:2006fg ;
Boyarsky:2006ag ; Boyarsky:2007ay ; Yuksel:2007xh ; Seljak:2006qw ;
Boyarsky:2008xj ; Perez:2016tcq . Meanwhile, more complex mechanisms for the
production of DM, such as the Shi-Fuller mechanism Shi:1998km  which describes
resonant oscillation production or other non-thermal production mechanisms
including the decay of an extra-singlet scalar Shaposhnikov:2006xi ;
Kusenko:2006rh , and scatterings through new mediators in the thermal bath
without reaching thermal equilibrium Khalil:2008kp ; Kaneta:2016vkq ;
Biswas:2016bfo ; Seto:2020udg , have been proposed.

Before ending this section, we emphasize that if the production mechanism of
the sterile neutrino DM is based on the DW mechanism, the area excluded by the
results of the CB effect is placed in the region that was already ruled out by
the X-ray experiment. However, if sterile neutrinos are produced through other
mechanisms, masses less than 1 KeV and mixing angles larger than $10^{-4}$ are
allowed; therefore, the constraint obtained from the results of the CB effect
is a new constraint and excludes part of those areas.

As a final point, it is worth mentioning that, here, we considered sterile
neutrino DM in the context of seesaw model. However, it is possible to
introduce the right-handed sterile neutrinos as the DM candidates which can be
coupled effectively to the SM particles through the right-handed current
interactions with the SM intermediate gauge bosons Xue:2016dpl ; Xue:2015wha ;
Xue:1997tz ; Xue:1996fm . Indeed, this model was motivated by the parity
symmetry reconstruction at high energies without any extra gauge bosons. This
sort of DM candidates might also be considered as a new source of CB which is
under investigation as a future work.

##  V  Conclusion

We examined whether the existence of the CB angle of CMB photons could be used
as a tool to study the nature and properties of DM. To this end, we considered
two types of DM candidates, i.e. dipolar and sterile neutrino DM, and
calculated the forward scattering contribution to investigate the CMB
polarization effects on DM properties. We found that the interaction of those
probable DM candidates and CMB photons results in generating the B mode
polarization patterns and, consequently, produces the CMB CB effect. Using the
birefringence angle reported by using the Planck data release, we discussed
the properties of the mentioned candidates of DM, and the results are as
follows:

  * •

Calculations performed regarding the dipolar DM showed that this DM candidate
can contribute to generate a part of the CB effect of the CMB. Moreover, from
another point of view, we used the reported CB angle of the CMB to put a new
constraint on its electromagnetic coupling and mass, i.e.,
$\mathcal{M}/10^{-15}e.cm\approx 10\,\sqrt{m_{DM}/GeV}$, which means that for
dipolar DM particles whose mass is about $m_{\text{\tiny{DM}}}\approx 3\,MeV$,
the magnetic dipole moment will be around $\mathcal{M}\leqslant(7.6\pm
1.1)\times 10^{-16}\,e\text{\,cm}$. For further clarification, we provided
Fig.2 in which the full mass dependence of the result has been illustrated.
Note that this figure is originally adapted from refSigurdson:2004zp  and we
added our result to it. Indeed, we put a constraint on the sub-GeV Majorana
dipolar DM through the CB effect of the CMB photons. It is also notable that
the constraints obtained on Majorana dipolar DM, including the constraints
that come from the direct detection experiments Masso:2009mu , study DM
particles whose mass is around GeV and higher, while the constraint that we
obtain here is concerning the sub-GeV DM particles.

  * •

In the case of the sterile neutrino DM, we found that this sort of DM
candidate can also contribute to produce a part of the CB angle of the CMB.
Furthermore, by using the reported CB angle of the CMB, the mixing angle
$\theta^{2}$ and mass of the sterile neutrino are constrained as
$\theta^{2}\approx(3.3\pm 1.1)\,\,(rad)^{2}\,\frac{m_{DM}}{KeV}$. It seems
that this constraint has already been excluded by some experiments such as
X-ray experiment, which explains the exclusion based on the DW mechanism of
sterile neutrino production. However, it should be pointed out that the values
of mass ($m_{\text{\tiny{DM}}}$) and the mixing angle ($\theta$) could depend
on the cosmological production scenarios. Besides the DW mechanism, more
complex mechanisms for the production of DM, such as the Shi-Fuller mechanism
or other non-thermal production mechanisms including the decay of an extra
singlet scalar, and scatterings through new mediators in the thermal bath
without reaching thermal equilibrium, have been proposed. It is notable that
the importance of our result depends on the sterile neutrino production
mechanism; if the production mechanism of the sterile neutrino DM is based on
the DW mechanism, the area which is excluded by the results of the CB effect
is placed in the region that already been ruled out by the X-ray experiment.
Meanwhile, if sterile neutrinos are produced through other mechanisms, masses
less than 1 KeV and mixing angles larger than $10^{-4}$ are allowed and,
therefore, the constraint obtained from the results of the CB effect is a new
constraint and excludes part of those areas.

## Acknowledgment

S. Mahmoudi is grateful to the Iran Science Elites Federation for the
financial support.

##  Appendix A Studying the contribution of the photon-dipolar DM scattering
on the CMB polarization in case ${\Large{\delta\gg k^{0}}}$

In this appendix, we are going to calculate the contribution of the photon-
Majorana dipolar DM forward scattering in case ${{\delta\gg k^{0}}}$. In this
case, Eq. (23) can be estimated as follows

$\displaystyle{M}\simeq\,\mathcal{M}_{12}^{2}\,\,\frac{(2k\cdot
q)^{2}}{(m_{\text{\tiny{DM}}_{2}}^{2}-m_{\text{\tiny{DM}}_{1}}^{2})^{2}}\,\,{\bf{k}}\cdot(\vec{\epsilon}_{s^{\prime}}\times\vec{\epsilon}_{s})+\,(1\,\leftrightarrow\,2).$
(67)

Working in the non-relativistic limit (${\bf{q}}\approx m_{\text{\tiny{DM}}}$)
and assuming $m_{\text{\tiny{DM}}_{1}}$ to be the same order of
$m_{\text{\tiny{DM}}_{2}}$, one will arrive at the following relation

$\displaystyle{M}\simeq\,\mathcal{M}_{12}^{2}\,\,(\frac{k^{0}}{\delta})^{2}\,\,{\bf{k}}\cdot(\vec{\epsilon}_{s^{\prime}}\times\vec{\epsilon}_{s})\,+\,(1\,\leftrightarrow\,2).$
(68)

For the cases in which $k^{0}\ll\delta\ll m_{\text{\tiny{DM}}}$,
$(m_{\text{\tiny{DM}}_{1}}\approx m_{\text{\tiny{DM}}_{2}}\approx
m_{\text{\tiny{DM}}})$, and after some calculation, one can find the evolution
of the Stokes parameters similar to (28)-(30) except that
$\dot{\tau}_{\text{\tiny{DM}}}$ is defined as follows

$\dot{\tau}_{\text{\tiny{DM}}}=\,(\frac{k^{0}}{\delta})^{2}\frac{\mathcal{M}^{2}}{m_{\text{\tiny{DM}}}}\,\rho_{\text{\tiny{DM}}}.$
(69)

The above relation clearly shows that the contribution of the photon-dipolar
DM scattering on the CMB polarization will be suppressed as
$(\frac{k^{0}}{\delta})^{2}$ compared to the case $\delta\ll k^{0}$ and
therefore, we will not consider this case in this paper.

##  Appendix B Calculation of Time evolution of the density matrix components
via photon-Sterile neutrino interaction

This appendix aims to calculate the time evolution of the density matrix
components due to the forward scattering of the photon-Sterile neutrino
interaction. To this end, we use the seesaw Lagrangian given in (57) to find
the types of possible interactions between sterile neutrinos and photons.
Indeed, the dominant interaction comes from the scattering of photons from
Sterile neutrinos at a one-loop level with a lepton and weak gauge bosons
propagating in the loop. Representative relevant Feynman diagrams are shown in
Fig. 3.

Fourier transformations of the electromagnetic free gauge field $A^{\mu}$ and
Majorana fermion field $N(x)$, which are self-conjugate, are as follows

$A_{\mu}(x)=\int\frac{d^{3}{\bf
k}}{(2\pi)^{3}2k^{0}}[a_{s}(p)\epsilon_{s\mu}(k)e^{-ik.x}+a_{s}^{\dagger}(k)\epsilon_{s\mu}^{*}(k)e^{ik.x}],$
(70) $N(x)=\int\frac{d^{3}{\bf
q}}{(2\pi)^{3}}\frac{m_{\tiny{\text{DM}}}}{q^{0}}\left[b_{r}(q)u_{r}(q)e^{-iq\cdot
x}+b_{r}^{\dagger}(q)v_{r}(q)e^{iq\cdot x}\right],$ (71)

where $\epsilon_{s\mu}(p)$ with $s=1,2$ are the photon polarization 4-vectors
of two physical transverse polarization while $u_{r}(q)$ and $v_{r}(q)$ are
the Dirac spinors. The creation $a_{s}^{\dagger}(k)$ ($b_{r}^{\dagger}(q)$)
and annihilation $a_{s}(k)$ ($b_{r}(q)$) operators respect the following
canonical commutation (anti-commutation) relations

$\displaystyle[a_{s}(k),a_{s^{\prime}}^{\dagger}(k^{\prime})]$
$\displaystyle=$
$\displaystyle(2\pi)^{3}2k^{0}\delta_{ss^{\prime}}\delta^{(3)}({\bf k}-{\bf
k^{\prime}}),$
$\displaystyle\\{b_{r}(q),b_{r^{\prime}}^{\dagger}(q^{\prime})\\}$
$\displaystyle=$
$\displaystyle(2\pi)^{3}\frac{q^{0}}{m_{\tiny{\text{DM}}}}\delta_{rr^{\prime}}\delta^{(3)}({\bf
q}-{\bf q^{\prime}}).$ (72)

Making use of Eqs. (3), (57), and the above relations, one can find that the
leading-order interacting Hamiltonian for the scattering representing in Fig.
3 can be expressed by the following scattering amplitude

$\displaystyle H_{I}^{0}(t)$ $\displaystyle=$
$\displaystyle\,\int{d\bf{q}}{d\bf{q^{\prime}}}{d\bf{k}}{d\bf{k^{\prime}}}(2\pi)^{3}\delta^{(3)}({\bf{q^{\prime}}}+{\bf{k^{\prime}}}-{\bf{q}}-{\bf{k}})\exp({i[q^{\prime
0}+k^{\prime 0}-q^{0}-k^{0}]})$ (73) $\displaystyle\times$
$\displaystyle[b_{r^{\prime}}^{\dagger}({\bf
q^{\prime}})a_{s^{\prime}}^{\dagger}({\bf
k^{\prime}})\mathcal{M}_{\text{tot}}(N\gamma\,\to\,N\gamma)\,a_{s}({\bf
k})b_{r}({\bf q})],$

with
${d\bf{q}}\equiv\frac{d^{3}\bf{q}}{(2\pi)^{3}}\frac{\tiny{\text{DM}}}{q^{0}}$,
${d\bf{k}}\equiv\frac{d^{3}\bf{k}}{(2\pi)^{3}}\frac{1}{2k^{0}}$ and the total
amplitude ${M}_{\text{tot}}$ can be obtained from the sum of all Feynman
diagrams in Fig.3, as follows

$\displaystyle{M}_{tot}({\bf q^{\prime}}r^{\prime},{\bf
k^{\prime}}s^{\prime},{\bf q}r,{\bf k}s)$ $\displaystyle\equiv$
$\displaystyle{M}_{1}({\bf q^{\prime}}r^{\prime},{\bf
k^{\prime}}s^{\prime},{\bf q}r,{\bf k}s)+{M}_{2}({\bf
q^{\prime}}r^{\prime},{\bf k^{\prime}}s^{\prime},{\bf q}r,{\bf k}s)$ (74)
$\displaystyle-{M}_{3}({\bf q^{\prime}}r^{\prime},{\bf
k^{\prime}}s^{\prime},{\bf q}r,{\bf k}s)-{M}_{4}({\bf
q^{\prime}}r^{\prime},{\bf k^{\prime}}s^{\prime},{\bf q}r,{\bf k}s),$

where ${M}_{3,4}({\bf q^{\prime}}r^{\prime},{\bf k^{\prime}}s^{\prime},{\bf
q}r,{\bf k}s)$ are, respectively, the Hermitian conjugates of ${M}_{1,2}({\bf
q^{\prime}}r^{\prime},{\bf k^{\prime}}s^{\prime},{\bf q}r,{\bf k}s)$ and have
been contributed from antiparticles in the loops as follows

$\displaystyle{M}_{1}({\bf q^{\prime}}r^{\prime},{\bf
k^{\prime}}s^{\prime},{\bf q}r,{\bf k}s)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{4}}\,\frac{e^{2}\,g^{2}}{8}\,\theta^{2}\int
d^{4}l\,\,\,\bar{u}_{r^{{}^{\prime}}}({\bf
q^{{}^{\prime}}})\gamma^{\alpha}\,(1-\gamma^{5})\,S_{F}(l+k-k^{{}^{\prime}})\not{\epsilon}_{s^{{}^{\prime}}}({\bf
k^{{}^{\prime}}})$ (75) $\displaystyle S_{F}(k+l)\,\not{\epsilon}_{s}({\bf
k})\,S_{F}(l)\gamma^{\beta}\,(1-\gamma^{5})\,u_{r}({\bf
q})\,D_{F_{\alpha\beta}}(q-l),$ $\displaystyle{M}_{2}({\bf
q^{\prime}}r^{\prime},{\bf k^{\prime}}s^{\prime},{\bf q}r,{\bf k}s)$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{4}}\,\frac{e^{2}\,g^{2}}{8}\,\theta^{2}\int
d^{4}l\,\,\,\bar{u}_{r^{{}^{\prime}}}({\bf
q^{{}^{\prime}}})\gamma^{\alpha}\,(1-\gamma^{5})\,S_{F}(l+k-k^{{}^{\prime}})\not{\epsilon}_{s}({\bf
k})$ (76) $\displaystyle
S_{F}(l-k^{{}^{\prime}})\,\not{\epsilon}_{s^{{}^{\prime}}}({\bf
k^{{}^{\prime}}})\,S_{F}(l)\gamma^{\beta}\,(1-\gamma^{5})\,u_{r}({\bf
q})\,D_{F_{\alpha\beta}}(q-l),$ $\displaystyle{M}_{3}({\bf
q^{\prime}}r^{\prime},{\bf k^{\prime}}s^{\prime},{\bf q}r,{\bf k}s)$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{4}}\,\frac{e^{2}\,g^{2}}{8}\,\theta^{2}\int
d^{4}l\,\,\,\bar{v}_{r}({\bf
q})\gamma^{\alpha}\,(1+\gamma^{5})\,S_{F}(-l)\not{\epsilon}_{s}({\bf
k})S_{F}(-k-l)$ (77) $\displaystyle\,\not{\epsilon}_{s^{{}^{\prime}}}({\bf
k^{{}^{\prime}}})\,S_{F}(k^{{}^{\prime}}-k-l)\gamma^{\beta}\,(1+\gamma^{5})\,v_{r}({\bf
q^{\prime}})\,D_{F_{\alpha\beta}}(l-q),$

and

$\displaystyle{M}_{4}({\bf q^{\prime}}r^{\prime},{\bf
k^{\prime}}s^{\prime},{\bf q}r,{\bf k}s)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{4}}\,\frac{e^{2}\,g^{2}}{8}\,\theta^{2}\int
d^{4}l\,\,\,\bar{v}_{r}({\bf
q})\gamma^{\alpha}\,(1+\gamma^{5})\,S_{F}(-l)\not{\epsilon}_{s^{{}^{\prime}}}({\bf
k^{{}^{\prime}}})S_{F}(k^{{}^{\prime}}-l)$ (78)
$\displaystyle\,\not{\epsilon}_{s}({\bf
k})\,S_{F}(k^{{}^{\prime}}-k-l)\gamma^{\beta}\,(1+\gamma^{5})\,v_{r^{{}^{\prime}}}({\bf
q^{{}^{\prime}}})\,D_{F_{\alpha\beta}}(l-q),$

where $S_{F}$ denotes the fermion propagator, the indices $r,r^{\prime}$ and
$s,s^{\prime}$ stand for the Sterile neutrino and photon spin states,
respectively. Now, in order to calculate the forward scattering term in (2),
one should find the commutator $[H_{I}^{0}(t),D_{ij}^{0}({\bf p})]$, then
evaluate the expectation value $\langle[H_{I}^{0}(t),D_{ij}^{0}({\bf
p})]\rangle$ according to the following operator expectation value

$\langle\,b^{\dagger}_{r^{\prime}_{i}}(q^{\prime})b_{r_{j}}(q)\,\rangle=(2\pi)^{3}\delta^{3}(\mathbf{q}-\mathbf{q^{\prime}})\delta_{rr^{\prime}}\delta_{ij}\frac{1}{2}f_{\text{\tiny{DM}}}(\mathbf{x},\mathbf{q}).$
(79)

In this regard, one can substitute (74-78) into (73) and then (2) to find the
time evolution of the density matrix components which is obtained as follows

$\displaystyle\frac{d}{dt}\rho_{ij}(k)\,$ $\displaystyle=$
$\displaystyle\,-\frac{\sqrt{2}}{12\,\pi\,k^{0}}\alpha\,\theta^{2}\,{G}_{\text{\tiny{F}}}\int
d{\bf{q}}\,\,(\delta_{is}\rho_{s^{\prime}j}(k)-\delta_{js^{\prime}}\rho_{is}(k))\,f_{\text{\tiny{DM}}}({\bf{x}},{\bf{q}})\,\bar{u}_{r}(q)\,\,(1-\gamma^{5})$
(80)
$\displaystyle(q\cdot\epsilon_{s}\,\,\not{\epsilon}_{s^{{}^{\prime}}}\,+\,q\cdot\epsilon_{s^{{}^{\prime}}}\,\,\not{\epsilon}_{s})\,u_{r}(q)+\frac{\sqrt{2}}{24\,\pi\,k^{0}}\alpha\,\theta^{2}\,{G}_{\text{\tiny{F}}}\int
d{\bf{q}}\,\,(\delta_{is}\rho_{s^{\prime}j}(k)-\delta_{js^{\prime}}\rho_{is}(k))\,$
$\displaystyle
f_{\text{\tiny{DM}}}({\bf{x}},{\bf{q}})\,\bar{u}_{r}(q)\,(1-\gamma^{5})\,\not{k}\,(\not{\epsilon}_{s^{{}^{\prime}}}\,\not{\epsilon}_{s}\,-\,\not{\epsilon}_{s}\,\not{\epsilon_{s^{{}^{\prime}}}})\,u_{r}(q).$

## References

  * (1) M.S. Roberts and R.N. Whitehurst, ”The rotation curve and geometry of M31 at large galactocentric distances”, Astrophys. J. 201, 327 (1975).
  * (2) S. D. M. White and M. J. Rees, “Core condensation in heavy halos: A Two stage theory for galaxy formation and clusters,” Mon. Not. Roy. Astron. Soc. 183, 341-358 (1978)
  * (3) D. N. Spergel et al. [WMAP], “First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Determination of cosmological parameters,” Astrophys. J. Suppl. 148, 175-194 (2003)
  * (4) D. Clowe, A. Gonzalez and M. Markevitch, “Weak lensing mass reconstruction of the interacting cluster 1E0657-558: Direct evidence for the existence of dark matter,” Astrophys. J. 604, 596-603 (2004)
  * (5) Y. Cai, T. Han, T. Li and R. Ruiz, “Lepton Number Violation: Seesaw Models and Their Collider Tests,” Front. in Phys. 6, 40 (2018)
  * (6) H. Baer, V. Barger, P. Huang and X. Tata, “Natural Supersymmetry: LHC, dark matter and ILC searches,” JHEP 05, 109 (2012)
  * (7) K. Sigurdson, M. Doran, A. Kurylov, R. R. Caldwell and M. Kamionkowski, “Dark-matter electric and magnetic dipole moments,” Phys. Rev. D 70, 083501 (2004) [erratum: Phys. Rev. D 73, 089903 (2006)]
  * (8) J. Kumar, D. Marfatia and D. Yaylali, “Vector dark matter at the LHC,” Phys. Rev. D 92, no.9, 095027 (2015)
  * (9) R. Catena, K. Fridell and V. Zema, “Direct detection of fermionic and vector dark matter with polarised targets,” JCAP 11, 018 (2018)
  * (10) P. W. Graham and S. Rajendran, “New Observables for Direct Detection of Axion Dark Matter,” Phys. Rev. D 88, 035023 (2013)
  * (11) M. Duerr, P. Fileviez Pérez and J. Smirnov, “Scalar Dark Matter: Direct vs. Indirect Detection,” JHEP 06, 152 (2016)
  * (12) C. A. Arguelles and J. Kopp, “Sterile neutrinos and indirect dark matter searches in IceCube,” JCAP 07, 016 (2012)
  * (13) J. W. Foster, “Direct and Indirect Searches for Axion Dark Matter,”
  * (14) R. Ramos, Van Que Tran and T. C. Yuan, Phys. Rev. D 103, no.7, 075021 (2021)
  * (15) S. Mahmoudi, M. Haghighat, S. A. M. Vamegh and R. Mohammadi, “Dipolar dark matter and CMB B-mode polarization,” Eur. Phys. J. C 80, no.5, 402 (2020)
  * (16) S. Modares Vamegh, M. Haghighat, S. Mahmoudi and R. Mohammadi, “Impact of the vector dark matter on polarization of the CMB photon,” Phys. Rev. D 100, no.10, 103024 (2019)
  * (17) M. Haghighat, S. Mahmoudi, R. Mohammadi, S. Tizchang and S. S. Xue, “Circular polarization of cosmic photons due to their interactions with Sterile neutrino dark matter,” Phys. Rev. D 101, no.12, 123016 (2020)
  * (18) A. M. Green, “Dark matter in astrophysics/cosmology,” SciPost Phys. Lect. Notes 37, 1 (2022)
  * (19) J. Cooley, “Dark Matter Direct Detection of Classical WIMPs,” [arXiv:2110.02359 [hep-ph]].
  * (20) T. R. Slatyer, “Les Houches Lectures on Indirect Detection of Dark Matter,” [arXiv:2109.02696 [hep-ph]].
  * (21) A. Arvanitaki, M. Baryakhtar and X. Huang, “Discovering the QCD Axion with Black Holes and Gravitational Waves,” Phys. Rev. D 91, no.8, 084011 (2015)
  * (22) V. Cardoso, Ó. J. C. Dias, G. S. Hartnett, M. Middleton, P. Pani and J. E. Santos, “Constraining the mass of dark photons and axion-like particles through black-hole superradiance,” JCAP 03, 043 (2018)
  * (23) M. J. Stott and D. J. E. Marsh, “Black hole spin constraints on the mass spectrum and number of axionlike fields,” Phys. Rev. D 98, no.8, 083006 (2018)
  * (24) F. Hofmann and C. Wegg, “7.1 keV sterile neutrino dark matter constraints from a deep Chandra X-ray observation of the Galactic bulge Limiting Window,” Astron. Astrophys. 625, L7 (2019)
  * (25) C. W. Chiang and T. Yamada, “3.5-keV X-ray line from nearly-degenerate WIMP dark matter decays,” JHEP 09, 006 (2014)
  * (26) D. Cadamuro and J. Redondo, “Cosmological bounds on pseudo Nambu-Goldstone bosons,” JCAP 02, 032 (2012)
  * (27) P. Arias, D. Cadamuro, M. Goodsell, J. Jaeckel, J. Redondo and A. Ringwald, “WISPy Cold Dark Matter,” JCAP 06, 013 (2012)
  * (28) T. Higaki, N. Kitajima and F. Takahashi, “Hidden axion dark matter decaying through mixing with QCD axion and the 3.5 keV X-ray line,” JCAP 12, 004 (2014)
  * (29) M. Pospelov, A. Ritz, C. Skordis, A. Ritz and C. Skordis, “Pseudoscalar perturbations and polarization of the cosmic microwave background,” Phys. Rev. Lett. 103, 051302 (2009)
  * (30) M. A. Fedderke, P. W. Graham and S. Rajendran, “Axion Dark Matter Detection with CMB Polarization,” Phys. Rev. D 100, no.1, 015040 (2019)
  * (31) P. Agrawal, A. Hook and J. Huang, “A CMB Millikan experiment with cosmic axiverse strings,” JHEP 07, 138 (2020)
  * (32) A. Lue, L. M. Wang and M. Kamionkowski, “Cosmological signature of new parity violating interactions,” Phys. Rev. Lett. 83, 1506-1509 (1999)
  * (33) T. Fujita, K. Murai, H. Nakatsuka and S. Tsujikawa, “Detection of isotropic cosmic birefringence and its implications for axionlike particles including dark energy,” Phys. Rev. D 103, no.4, 043509 (2021)
  * (34) G. C. Liu, S. Lee and K. W. Ng, “Effect on cosmic microwave background polarization of coupling of quintessence to pseudoscalar formed from the electromagnetic field and its dual,” Phys. Rev. Lett. 97, 161303 (2006)
  * (35) S. Gasparotto and I. Obata, “Cosmic Birefringence from Monodromic Axion Dark Energy,” [arXiv:2203.09409 [astro-ph.CO]].
  * (36) Y. Minami and E. Komatsu, “New Extraction of the Cosmic Birefringence from the Planck 2018 Polarization Data,” Phys. Rev. Lett. 125, no.22, 221301 (2020)
  * (37) E. Komatsu et al. [WMAP], “Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation,” Astrophys. J. Suppl. 180, 330-376 (2009)
  * (38) P. A. R. Ade et al. [Planck], “Planck 2015 results - II. Low Frequency Instrument data processings,” Astron. Astrophys. 594, A2 (2016)
  * (39) T. Namikawa, Y. Guan, O. Darwish, B. D. Sherwin, S. Aiola, N. Battaglia, J. A. Beall, D. T. Becker, J. R. Bond and E. Calabrese, et al. “Atacama Cosmology Telescope: Constraints on cosmic birefringence,” Phys. Rev. D 101, no.8, 083527 (2020)
  * (40) S. K. Choi et al. [ACT], “The Atacama Cosmology Telescope: a measurement of the Cosmic Microwave Background power spectra at 98 and 150 GHz,” JCAP 12, 045 (2020)
  * (41) E. Komatsu, “New physics from the polarized light of the cosmic microwave background,” Nature Rev. Phys. 4, no.7, 452-469 (2022)
  * (42) Y. Minami and E. Komatsu, “Simultaneous determination of the cosmic birefringence and miscalibrated polarization angles II: Including cross frequency spectra,” PTEP 2020, no.10, 103E02 (2020)
  * (43) A. Abghari, R. M. Sullivan, L. T. Hergt and D. Scott, “Constraints on cosmic birefringence using $E$-mode polarisation,” [arXiv:2203.10733 [astro-ph.CO]].
  * (44) R. Mohammadi, J. Khodagholizadeh, M. Sadegh and A. Vahedi, “Cross-correlation Power Spectra and Cosmic Birefringence of the CMB via Photon-neutrino Interaction,” [arXiv:2109.00152 [hep-ph]].
  * (45) J. Khodagholizadeh, R. Mohammadi and S. S. Xue, “Photon-neutrino scattering and the B-mode spectrum of CMB photons,” Phys. Rev. D 90 (2014) no.9, 091301 [arXiv:1406.6213 [astro-ph.CO]].
  * (46) R. Mohammadi, J. Khodagholizadeh, M. Sadegh and S. S. Xue, “B-mode polarization of the CMB and the cosmic neutrino background,” Phys. Rev. D 93 (2016) no.12, 125029 [arXiv:1602.00237 [astro-ph.CO]].
  * (47) J. Khodagholizadeh, R. Mohammadi, M. Sadegh and A. Vahedi, “B-mode Power Spectrum of CMB via Polarized Compton Scattering,” JCAP 01 (2020), 051 [arXiv:1909.00568 [astro-ph.CO]].
  * (48) S. Tizchang, S. Batebi, M. Haghighat and R. Mohammadi, “Cosmic microwave background polarization in non-commutative space-time,” Eur. Phys. J. C 76 (2016) no.9, 478 [arXiv:1605.09045 [hep-ph]].
  * (49) J. Khodagholizadeh, R. Mohammadi, S. M. S. Movahed, “ CMB Polarization by the Asyemmetric Template of Scalar Perturbation,” Eur. Phys. J. C 83, 651 (2023).
  * (50) A. Kosowsky, “Cosmic microwave background polarization,” Annals Phys. 246 (1996), 49-85
  * (51) J. R. Bond and G. Efstathiou, “Cosmic background radiation anisotropies in universes dominated by nonbaryonic dark matter,” Astrophys. J. Lett. 285 (1984), L45-L48
  * (52) M. Zaldarriaga and U. Seljak, Phys. Rev. D 55, 1830 (1997)
  * (53) P. Diego-Palazuelos, J. R. Eskilt, Y. Minami, M. Tristram, R. M. Sullivan, A. J. Banday, R. B. Barreiro, H. K. Eriksen, K. M. Górski and R. Keskitalo, et al. “Cosmic Birefringence from the Planck Data Release 4,” Phys. Rev. Lett. 128, no.9, 091302 (2022)
  * (54) U. Seljak and M. Zaldarriaga, “A Line of sight integration approach to cosmic microwave background anisotropies,” Astrophys. J. 469, 437 (1996)
  * (55) E. Masso, S. Mohanty and S. Rao, “Dipolar Dark Matter,” Phys. Rev. D 80, 036009 (2009)
  * (56) H. M. Lee, “Magnetic dark matter for the X-ray line at 3.55 keV,” Phys. Lett. B 738, 118-122 (2014)
  * (57) S. Davidson, M. Gorbahn and A. Santamaria, “From transition magnetic moments to majorana neutrino masses,” Phys. Lett. B 626, 151-160 (2005)
  * (58) A. Kosowsky, “Cosmic microwave background polarization,” Annals Phys. 246, 49 (1996)
  * (59) S. Dodelson, “Modern Cosmology,” Academic Press, 2003, ISBN 978-0-12-219141-1
  * (60) N. Aghanim et al. [Planck], “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641, A6 (2020) [erratum: Astron. Astrophys. 652, C4 (2021)]
  * (61) S. Burles, K. M. Nollett, J. W. Truran and M. S. Turner, “Sharpening the predictions of big bang nucleosynthesis,” Phys. Rev. Lett. 82, 4176 (1999)
  * (62) A. Boyarsky, M. Drewes, T. Lasserre, S. Mertens and O. Ruchayskiy, “Sterile neutrino Dark Matter,” Prog. Part. Nucl. Phys. 104, 1 (2019)
  * (63) A. D. Dolgov, “Neutrinos in cosmology,” Phys. Rept. 370, 333 (2002)
  * (64) F. Bezrukov, A. Chudaykin and D. Gorbunov, “Hiding an elephant: heavy Sterile neutrino with large mixing angle does not contradict cosmology,” JCAP 1706, no. 06, 051 (2017)
  * (65) S. Dodelson and L. M. Widrow, “Sterile-neutrinos as dark matter,” Phys. Rev. Lett. 72, 17-20 (1994)
  * (66) J. Buckley, D. F. Cowen, S. Profumo, A. Archer, M. Cahill-Rowley, R. Cotta, S. Digel, A. Drlica-Wagner, F. Ferrer and S. Funk, et al. “Working Group Report: WIMP Dark Matter Indirect Detection,” [arXiv:1310.7040 [astro-ph.HE]].
  * (67) A. Boyarsky, A. Neronov, O. Ruchayskiy and M. Shaposhnikov, “Constraints on sterile neutrino as a dark matter candidate from the diffuse x-ray background,” Mon. Not. Roy. Astron. Soc. 370, 213-218 (2006).
  * (68) A. Boyarsky, A. Neronov, O. Ruchayskiy, M. Shaposhnikov and I. Tkachev, “Where to find a dark matter sterile neutrino?,” Phys. Rev. Lett. 97, 261302 (2006).
  * (69) A. Boyarsky, J. Nevalainen and O. Ruchayskiy, “Constraints on the parameters of radiatively decaying dark matter from the dark matter halo of the Milky Way and Ursa Minor,” Astron. Astrophys. 471, 51-57 (2007).
  * (70) A. Boyarsky, D. Iakubovskyi, O. Ruchayskiy and V. Savchenko, “Constraints on decaying Dark Matter from XMM-Newton observations of M31,” Mon. Not. Roy. Astron. Soc. 387, 1361 (2008).
  * (71) H. Yuksel, J. F. Beacom and C. R. Watson, “Strong Upper Limits on Sterile Neutrino Warm Dark Matter,” Phys. Rev. Lett. 101, 121301 (2008).
  * (72) U. Seljak, A. Makarov, P. McDonald and H. Trac, “Can sterile neutrinos be the dark matter?,” Phys. Rev. Lett. 97, 191303 (2006).
  * (73) A. Boyarsky, J. Lesgourgues, O. Ruchayskiy and M. Viel, “Lyman-alpha constraints on warm and on warm-plus-cold dark matter models,” JCAP 05, 012 (2009).
  * (74) K. Perez, K. C. Y. Ng, J. F. Beacom, C. Hersh, S. Horiuchi and R. Krivonos, “Almost closing the $\nu$MSM sterile neutrino dark matter window with NuSTAR,” Phys. Rev. D 95, no.12, 123002 (2017).
  * (75) X. D. Shi and G. M. Fuller, “A New dark matter candidate: Nonthermal sterile neutrinos,” Phys. Rev. Lett. 82, 2832-2835 (1999)
  * (76) M. Shaposhnikov and I. Tkachev, “The nuMSM, inflation, and dark matter,” Phys. Lett. B 639, 414-417 (2006).
  * (77) A. Kusenko, “Sterile neutrinos, dark matter, and the pulsar velocities in models with a Higgs singlet,” Phys. Rev. Lett. 97, 241301 (2006).
  * (78) S. Khalil and O. Seto, “Sterile neutrino dark matter in B - L extension of the standard model and galactic 511-keV line,” JCAP 10, 024 (2008).
  * (79) K. Kaneta, Z. Kang and H. S. Lee, “Right-handed neutrino dark matter under the B$-$L gauge interaction,” JHEP 02, 031 (2017).
  * (80) A. Biswas and A. Gupta, “Freeze-in Production of Sterile Neutrino Dark Matter in U(1)B-L Model,” JCAP 09, 044 (2016).
  * (81) O. Seto and T. Shimomura, “Signal from sterile neutrino dark matter in extra $U(1)$ model at direct detection experiment,” Phys. Lett. B 811, 135880 (2020).
  * (82) S. S. Xue, “Hierarchy spectrum of SM fermions: from top quark to electron neutrino,” JHEP 1611, 072 (2016)
  * (83) S. S. Xue, “Neutrino masses and mixings,” Mod. Phys. Lett. A 14, 2701 (1999)
  * (84) S. S. Xue, “Vectorlike $W^{\pm}$-boson coupling at TeV and third family fermion masses,” Phys. Rev. D 93, no. 7, 073001 (2016)
  * (85) S. S. Xue, “Quark masses and mixing angles,” Phys. Lett. B 398, 177 (1997)