Dipolar dark matter and CMB B-mode polarization

We consider dark matter as singlet fermionic particles which carrying magnetic dipole moment to explore its contribution on the polarization of cosmic microwave background (CMB) photons. We show that Dirac fermionic dark matter has no contribution on the CMB polarization. However, in the case of Majorana dark matter this type of interaction leads to the B-mode polarization in presence of primordial scalar perturbations which is in contrast with standard scenario for the CMB polarization. We numerically calculate the B-mode power spectra and plot $C_l^{BB}$ for different dark matter masses and the $r$-parameter. We show that the dark matter with masses less than 100MeV have valuable contribution on $C_l^{BB}$. Meanwhile, the dark matters with mass $m_d\leq50MeV$ for $r=0.07$ ( $m_d\leq80MeV$ for $r=0.09$) can be excluded experimentally. Furthermore, our results put a bound on the magnetic dipole moment about $M\leq 10^{-16} e\,\,cm$ in agreement with the other reported constraints.


I. INTRODUCTION
The nature of dark matter (DM) and its interactions is one of the most important questions in cosmology and particle physics. In spite of the fact that there are wealth of cosmological evidence for existing dark matter, from galactic clusters and velocity curves of spining galaxies to gravitational lensing [1-3], its particle properties has remained elusive.
To explore the nature of dark matter, different experiments have been proposed such as DAMA/LIBRA collaboration at Gran Sasso [4], CoGeNT collaboration at the Soudan Laboratory underground [5] and CDMS collaboration [6] which are introduced to detect dark matter directly. In these experiments the scattering of dark matter from nucleons can be described by multipole interactions. In fact, a dark matter has zero electric charge and therefore in the simplest extension of standard model it can be coupled to photon through an intrinsic electric and or magnetic dipole moments which is well-known as dipolar dark matter (DDM) model [7][8][9][10][11]. However, the DDM model can successfully explain some claims of DAMA/LIBRA and COGENT collaborations [12,13].
The CMB photons is expected to be linearly polarized due to the anisotropic Compton scattering around the epoch of recombination. Meanwhile, according to the standard scenario of cosmology there is no physical mechanism to generate a circular polarized radiation at the last scattering surface. However, studies conducted in recent years show that the interaction between photon and matter can convert or generate the polarization states of photon in different situations. For instance, the linear polarization of the CMB photons can be converted to the circular polarization in the presence of background fields or due to the effects of particle scattering which has been widely discussed in the literature [14][15][16][17][18][19][20][21]. In this paper, we consider the DDM model with a singlet Majorana fermion as the dark matter to examine the effects of magnetic dipole moments on the CMB photon polarization.
Generally, the CMB polarization pattern has two geometrical components, E-mode and B-mode. These modes based on the Stokes parameters U and Q can form an independent local coordinate system [22][23][24][25][26]. According to the standard model of cosmology, while Emode polarization of Compton scattering at the last scattering surface can be produced due to the scalar and tensor perturbations, its B-mode polarization can only be produced by the tensor perturbations. Nevertheless, it has been shown that it is also possible to produce the B-mode polarization in the presence of scalar perturbations. Since the detection of B-mode polarization can provide a unique tool to investigate the CMB perturbations, it is important to identify all potential sources of the B-mode polarization. As the new sources, for instance, in [27] the effects of the Faraday rotation due to the uniform magnetic field on the CMB is investigated and it is shown that a nonvanishing B-mode can be produced through Farady rotation. In [28], the authors have discussed that photon-neutrino interaction in the presence of scalar perturbations could be considered as one of the sources of the CMB B-mode polarization. It is also shown that the Compton scattering in the non-commutative space time can generate the B-mode polarization of the CMB [29].
However, the parameter which characterizes the amplitude of metric tensor perturbation is r = P T /P S where P T = A T (k/k • ) n T −1 and P S = A S (k/k • ) n S −1 are, respectively, the power spectra of tensor and scalar metric perturbations and n T,S and A T,S are their spectral indices and amplitudes. The r parameter is usually calculated by comparing the B-mode and E-mode power spectra. Recent measurements of BICEP2 + Keck Array + Planck (BKP) report an upper bound r 0.002 < 0.09 [30].
In this work, we will show that magnetic like component of the CMB polarization (Bmode polarization) can be produced by the photon-DM interaction in the presence of scalar perturbations. The paper is organized as follows: we introduce the effective Lagrangian for the interaction of dipolar dark matter with photons in section 2. Then we give a brief introduction to the stokes parameters and drive the time evolution of these parameters in terms of the photon-DM scattering in section 3. The power spectrum is evaluated numerically in section 4. We compare our results with the experimental data and give some discussion in section 5.

II. DIPOLAR DARK MATTER MODEL
A particle as a candidate for the dark matter is generally known as a stable or relatively stable particle that does not interact electromagnetically. However in recent years, there are some interest in the study of electromagnetic interactions of DM. Such a particle has not probably the electric charge otherwise it has a significant interaction with the photons and could be easily detected. But this particle can weakly couple with the electromagnetic field through loop corrections. In fact, the most general form for the electromagnetic current between fermions consistent with the Lorentz covariance and the Ward identity can be written as follows [31]: where F 1 , G 1 , F 2 and G 2 are the electric, anapole, magnetic and electric dipole form factors, respectively. The current J em µ can be coupled with photons where its dipolar part is given by where F µν is the electromagnetic field, M and D are magnetic and electric dipole moment, respectively and σ µν = i 2 [γ µ , γ ν ]. The Lagrangian (2) forms the basis of the DDM model [7]. Therefore, the fermionic DM-particles electromagnetically interact with photons via electric and magnetic dipole moments. The polarization density matrix of photons is defined as where I is the total intensity of radiation, U, Q and V describe the polarization of photon and for unpolarized photon Q = U = V = 0. The circularly polarized radiation is defined by none-zero value for V and the linear polarization is described by the Stokes parameters Q and U. The parameters I and V are independent of reference frame whereas Q and U are frame dependent. Therefore, in the context of cosmology by introducing a set of linear combination of Q and U, one can find reference frame independent parameters that are known as E and B modes.
Meanwhile, time evolution of the Stokes parameters can be examined through the Boltzmann equation. This equation provides a systematic way to account for different couplings in a system and is generally expressed as follows where C[f ] in the right-hand side of (4) contains all possible collision terms while the lefthand side is known as the Liouville term and involves the effects of gravitational perturbations about the homogeneous cosmology. In the case of photon, the distribution function f is the density matrix ρ ij as is given in (3). Thus the density operator corresponding to the density matrix ρ ij can be given aŝ and the number operator D 0 , has an expectation value as fallows However, to examine the time evolution of the photons polarization in the CMB, we need the time evolution of the density matrix. To this end, We substitute (6) in where H is the full Hamiltonian, to find the time evolution of ρ ij as In (8) H 0 I is the interacting Hamiltonian at the lowest order [22]. The first and the second term on the right-handed side of (8) are called forward scattering and higher order collision term, respectively.
Here we consider a fermionic dark matter which interacts with photon via its magnetic dipole moment with the following Lagrangian FIG. 1: The typical diagrams for photon-dark matter scattering The Feynman diagram corresponding to DDM-photon scattering at the lowest order is very similar to the Compton scattering as is shown in Fig.1. Therefore, the interacting Hamiltonian at the lowest order can be obtained as follows with the Fourier transformations of the fields and propagator as follows and where ǫ sµ (k)'s are the photon polarization 4-vectors with s = 1, 2 for two physical transverse polarization of a free photon and a s (k)(a † s (k)) is the annihilation (creation) operator which satisfies the canonical commutation relation as In (12) U r and V r are the Dirac spinors, b r (d r ) and b † r (d † r ) are, respectively, the annihilation and creation operators for fermion (antifermion) satisfying Therefore, the interaction Hamiltonian (10) by using (11), (12) and (13) can be cast into with and Now, we are ready to evaluate the forward scattering term, the first term on the right hand side of (8). For this purpose, one needs the expectation value of operators such as [22] a 1 a 2 . and to evaluate [H 0 where or Meanwhile, in the nonrelativistic limit one has [32] where ξ is the two component spinor normalized to unity. Therefore, after some manipulations the amplitude can be rewritten as Eq. (28) for the Dirac fermions with both helicity degree of freedom (left and right handed helicity), leads to a vanishing average on the fermion helicity r as In fact, the photon-dark matter forward scattering, in the case of dark matter as a Dirac fermions with both handedness, has not any contribution on the CMB polarization. In contrast, for a dipolar fermionic dark matter as a singlet Majorana fermion, for instance, a right handed Majorana neutrino, using ξ † r σξ r = (−1) r+1 q/| q|, one has where v b = | q|/m f is the bulk velocity of dark matter [for example see [33]]. Although the second term in (30) is very similar to the first one, a straightforward calculation leads to a ignorable value for this term. In fact, in the second term the angle between ( ǫ s ′ × ǫ s ) and the bulk velocity is completely random. Note this term results dipole effect for CMB polarization which is difficult to determine from Doppler effects. Therefore, we ignore the second term of (30) for simplicity. Now by substituting (30) in (23) and using (8), the time evolution of density matrix element can be written as wherek = k/k 0 . Consequently, the stokes parameters evolve as where C I eγ ,C V eγ and C ± eγ show the effects of Thomson scattering [22] andτ d is defined as folloẇ where σ T is the Thomson cross section and the dark matter number density n d is It should be noted that the second term in the right hand side of (33) which comes from the photon-DDM forward scattering affects the time evolution of the stokes parameters Q and U.
To evaluate this term one needs the relation between magnetic dipole moment M and the dark matter scattering cross section σv [7] σv ≈ 1 2π which cast (35) intoτ where ρ d is the dark matter mass density. Since the number density of electron is equal to the number density of proton and it is approximately equal to the baryonic matter number density n e = n p ≈ n B.M then the ratio ofτ d with respect to theτ e corresponding to the Thomson cross section σ T can be found asτ where Ω d and Ω B.M are the dark matter density parameter and the baryonic matter density parameter, respectively. However, value of the ratio given in (40) can be estimated aṡ where for dark matter with masses 1GeV -1MeV varies as 5.2 × 10 −11 -5.2 × 10 −5 .
It should be noted that, we approximately consider v b ≈ 300 Km s −1 and for simplicity its dependence on the wave number and redshift is ignored. [see [33] and references there in].

IV. GENERALIZED BOLTZMANN EQUATION FOR THE CMB
The CMB polarization pattern includes two types of polarization, E and B-modes. While the E-mode polarization can be produced via scalar perturbations, the B-mode polarization is only generated by tensor perturbations. In the previous section we showed that the photon-DDM interaction can act as a source for generating B mode in the presence of scalar perturbations. The CMB radiation transfer is described by the multipole moments of temprature (I) and polarization (P ) [25,26] ∆ S I,P (η, K, µ) = ∞ l=0 (2l + 1)(−i) l ∆ S I,P l (η, K)P l (µ), where P l (µ) is the Legendre polynomial of rank l, µ =n.K = cos θ and θ is the angle between the CMB photon directionn = k |k| and the wave vectors K of the Fourier modes of scalar perturbations (S). Since for a given Fourier mode, one can choose a coordinate system in which K ẑ then the Boltzmann equation in the presence of Thomson scattering and DDM-photon interaction can be written as where Ψ and Φ are the metric perturbations, η is the conformal time, a(η) is the expansion factor which is normalized to unity for present time (η = η 0 ) and v b is the baryon bulk velocity, Π ≡ ∆ S 2 I + ∆ S 2 P − ∆ S• P and the polarization anisotropy is given by which can cast the equation of polarization anisotropy into [28] d dη Now by integrating (46) along the line of sight up to the present time η 0 , with the initial condition ∆ ±S P (0, K, µ) = 0, yields where x = K(η 0 − η) and or in terms of the redshift z To obtain (50) from (49), we have used the mass density of dark matter ρ d = ρ 0 d (1+z) 3 where ρ 0 d is mass density of dark matter in present time, adη = − dz H(1+z) and by using Friedmann equation in the matter dominated era it can be found as where H 0 ≈ 67Kms −1 Mpc −1 , Ω 0 M ≈ 0.31, Ω 0 Λ ≈ 0.69 [34]. Meanwhile E-mode and B-mode polarizations can be defined in terms of ∆ ±S P (η 0 , K, µ) as follows [22,[24][25][26] where ð andð are spin raising and lowering operators, respectively. Thus by assumming the scalar perturbation to be axially symmetric around K one has which can cast (52) and (53) into where g e (η) =τ e e −τe is the visibility function of electron. As one can easily see for τ d = 0 the equations (56) and (57) show that the DDM-photon interaction produces the nontrivial B-mode polarization and modify of the ordinary E-mode polarization. However, the power spectrum for the E and B-modes can be obtained by integrating over the initial power spectrum of the metric perturbation as [22,[24][25][26] C EE,S   Bl and modify the r-parameter as follows As (66) shows, the value of the r-parameter, as a scale of the amplitude of gravitational wave, is suppressed. By using (58-65) in (66), one can approximately find where r 0 is the standard tensor to scalar perturbation ratio without considering any new source for the B-mode polarization such as the CMB-DDM interaction.
However, the value of B-mode power spectrum in the presence of scalar perturbation and CMB-DDM interaction C