# Constraints on electromagnetic form factors of sub-GeV dark matter from the Cosmic Microwave Background anisotropy

Gaetano Lambiase, Subhendra Mohanty, Akhilesh Nautiyal, Soumya Rao

aa r X i v : . [ h e p - ph ] F e b Constraints on electromagnetic form factors of sub-GeV darkmatter from the Cosmic Microwave Background anisotropy

Gaetano Lambiase a , Subhendra Mohanty b , Akhilesh Nautiyal c and Soumya Rao ba Dipartimento di Fisica “E.R Caianiello”, Universit degli Studi di Salerno,Via Giovanni Paolo II, 132 - 84084 Fisciano (SA), Italy b Theory Division, Physical Research Laboratory,Navrangpura, Ahmedabad 380 009, India c Department of Physics, Malaviya National Instituteof Technology, JLN Marg, Jaipur 302017,India

Abstract

We consider dark matter which have non-zero electromagnetic form factors like electric/magneticdipole moments and anapole moment for fermionic dark matter and Rayleigh form factor forscalar dark matter. We consider dark matter mass m χ > O (MeV) and put constraints on theirmass and electromagnetic couplings from CMB and LSS observations. Fermionic dark matterwith non-zero electromagnetic form factors can annihilate to e + e − and scalar dark matter canannihilate to 2 γ at the time of recombination and distort the CMB. We analyze dark matterwith multipole moments with Planck and BAO observations. We ﬁnd upper bounds on anapolemoment g A < . × GeV -2 , electric dipole moment D < . × − e-cm, magnetic dipolemoment µ < . × − µ B , and the bound on Rayleigh form factor of dark matter is g / Λ < . × − GeV -2 with 95%C.L. PACS numbers: 98.80.Cq, 98.80.Ft, 26.35.+c, 95.35.+d . INTRODUCTION It is well accepted that formation of large scale structures and the rotation curves ofgalaxies require an extra dark matter component beyond than the known particles of thestandard model. The particle properties of this dark matter are, however, completely un-known. Direct detection experiments, which rely on nuclear scattering, have ruled out alarge parameter space. But, these techniques are not eﬃcient in measuring dark matter ofsub-GeV mass [1, 2]. To measure sub-GeV mass dark matter a suitable method is scatteringelectrons from heavy atoms [3–8]. Dark matter with non-zero electric or magnetic dipolemoments [9–11] or anapole moment [12, 13] can be very eﬀective in scattering electrons.The electromagnetic form factors can be viewed as eﬀective operators [14–17], which ariseby integrating out the heavy particles in a ultraviolet complete theory [18].The electromagnetic couplings of dark matter can be constrained from cosmic microwavebackground and large scale structures observations. The electric and magnetic dipolemoment vertex can give rise to dark matter-baryon coupling. For heavy dark matter( ∼

100 GeV) the baryon drag on the dark matter will show up in structure formation andCMB [10]. Light dark matter ( O (MeV) will annihilate to radiation and lower the eﬀectiveneutrino number ( N eﬀ ) [19–21].In this paper we will analyze the eﬀect of light dark matter with electromagnetic formfactors on CMB anisotropy and polarization from dark matter annihilation to e + e − or pho-tons close to recombination era, z ∼ e + , e − heats up the thermal gasand ionizes the neutral Hydrogen, which increases the free electron fraction. Due to thisincreased free electron fraction there is a broadening of the last scattering surface and sup-pression of CMB temperature anisotropy. The low- l correlations between polarization ﬂuc-tuations are also enhanced due to increased freeze-out value of the ionization fraction ofthe universe after recombination. These eﬀects on CMB are signiﬁcant and can be used toput constraints on thermal averaged annihilation cross-section h σv i . Planck-2018 reports h σv i < (3 . × − /f ) × ( M DM / (1GeV / c )) cm /s for velocity independent thermal aver-age cross-section [27]. Here f is the fraction of energy injected to the intergalactic medium(IGM) by annihilating dark matter.The annihilation χχ → e + e − occur with one dipole or anapole vertex and the annihilationcross sections are quadratic in dipole or anapole moments. For fermionic dark matter the χχ → γγ annihilation cross section are quartic in dipole moments and the bounds from thisprocess are much weaker [28] than the bounds derived from the CMB are much weaker thanthe ones we derive in this paper. For anapole dark matter the cross section for the process χχ → γγ is zero [12]. Scalar dark matter can have dimension-6 Rayleigh operator vertex withtwo-photons. For such Rayleigh dark matter the leading order contribution to annihilationwill be from the φφ → γγ process which can distort the CMB near recombination and fromthis we put bounds on the dark-matter photon Rayleigh coupling.This paper is organized as follows. In Section 2 we list the electromagnetic form factorsof dark matter which we shall constrain from CMB data. In Section 3 we discuss the physicsof recombination and the eﬀect of dark matter annihilation on the CMB. In Section 4 wecompare the CMB analysis with data from Planck and BAO and using COSMOMC we putconstraints on the dark matter form factors. In Section 5 we compare our bounds with2arlier results and give our conclusions.

2. ELECTROMAGNETIC FORM FACTORS OF DARK MATTER

Spin-1/2 dark matter can have the following electromagnetic form factors. These are themagnetic moment described by the dimension-5 operators, L magnetic = g Λ ¯ χσ µν χF µν (1)where g is a dimensionless coupling and Λ is the mass scale of the particles in the loopwhich generate the dipole moment. The magnetic moment of Dirac fermions is µ = 2 g / Λ and the operator (1) is zero for Majorana fermions.Similarly electric dipole operator is of dimension-5, L electric = g Λ i ¯ χσ µν γ χF µν (2)where the electric dipole moment of Dirac fermions is D = 2 g / Λ and the operator (2) iszero for Majorana fermions.Finally the anapole moment is a dimension-6 operator L anapole = g Λ i ¯ χγ µ γ χ∂ ν F µν (3)This operator is non-zero for Dirac as well as Majorana fermions and the coeﬃcient g A = g / (Λ ) is called the anapole moment of χ .Stringent bounds on sub-GeV mass dark matter are put from the observation of χe − → χe − scattering [3] in direct detection experiments like Xenon-10 [4], DarkSide [5] and Xenon-1T [6].Using the experimental limits on dark matter electron scattering from Xenon-10, Xenon-1T and DarkSide bound on the electric dipole, magnetic dipole and anapole form factors ofdark matter have been put in ref.[7, 8].Real and complex scalar dark matter can have interaction with 2-photons by dimension-6Rayleigh operator L φ γ = g Λ φ ∗ φF µν F µν (4)These will contribute to φφ → γγ annihilations which can be constrained from CMB [17].In the absence of CP violation the ˜ F F operator does not arise. The annihilation φφ → γγ takes place via s-wave in the leading order and cross section σ ( φφ → γγ ) v ≃ ( g ) m φ / Λ [17].Bounds on the operator 4 from Xenon1T [6]and gamma ray searches from dwarf spheroidalsatellites (dSphs) [29] and halo of the Milky way [30] by Fermi-LAT are obtained in ref. [17]for DM with mass larger than O (GeV). 3 . THERMAL HISTORY OF THE UNIVERSE WITH ANNIHILATING DARKMATTER Recombination occurs around z = 1100 when electrons and protons combine together toform neutral hydrogen. If the annihilation cross-section of dark matter particles is suﬃcientlylarge, it can modify the history of recombination and hence can leave a clear imprint onCMB power spectrum. The shower of particles produced due to annihilation can interactwith the thermal gas in three diﬀerent ways. ( i ) The annihilation products can ionize thethermal gas, ( ii ) can induce induce Lyman- α excitation of the hydrogen that will causemore electrons in n = 2 state and hence increase the ionization rate and ( iii ) can heat theplasma. Due to the ﬁrst two eﬀects the evolution of free electron fraction χ e changes andthe last eﬀect changes the temperature of baryons. The equation governing the evolution ofionization fraction in the presence of annihilating particles is given as dχ e dt = 1(1 + z ) H ( z ) [ R s ( z ) − I s ( z ) − I X ( z )] . (5)Here R s is the standard recombination rate, I s is the ionization rate due to standard sourcesand I X is the ionization rate due to annihilating dark matter particles. The computation ofstandard recombination rate was done in [31–33] and it is described as[ R s ( z ) − I s ( z )] = C × (cid:2) χ e n H α B − β B (1 − χ e ) e − h p ν s /k B T b (cid:3) . (6)Here n H is the number density of hydrogen nuclei, α B and β B are the eﬀective recombinationand photo-ionization rates for principle quantum numbers ≥ ν s is the frequency of the 2 s level from the ground state and T b is the temperature of the baryongas. The factor C appearing in Eqn. (6) is given by: C = [1 + K Λ s s n H (1 − χ e )][1 + K Λ s s n H (1 − χ e ) + Kβ B n H (1 − χ e )] . (7)Here Λ s s is the decay rate of the metastable 2 s level, n H (1 − χ e ) is the number of neutralground state H atoms and K = λ α πH ( z ) , where H ( z ) is the Hubble expansion rate at redshift z and λ α is the wavelength of the Ly − α transition from the 2 p level to the 1 s level.The term I X appearing in Eq. (5) represents the evolution of free electron density due tononstandard sources. In our case it is due to annihilation of dark matter during recombina-tion. which increases the ionization rate in two ways. ( i ) By direction ionization from theground state and ( ii ) by additional Ly − α photons, which boosts the population at n = 2increasing the the rate of photoionization by CMB. Hence, the ionization rate I X due todark matter annihilation is expressed as I X ( z ) = I Xi ( z ) + I Xα ( z ) . (8)Here I Xi ( z ) represents the ionization rate due to ionizing photons and I Xα represents theionization rate due to Ly − α photons.The rate of energy release dEdt per unit volume by a relic self-annihilating dark matterparticle can be expressed in terms of its thermally averaged annihilation cross-section h σv i m χ as dEdt = 2 gρ c c Ω DM (1 + z ) f ( z ) h σv i m χ , (9)where Ω DM is the dark matter density parameter, ρ c is the critical density today, g is degen-eracy factor 1 / / f ( z ) is the fractionof energy absorbed by the CMB plasma, which is O (1) factor and depends on redshift. Adetailed calculation of redshift dependence of f ( z ) for various annihilation channels is donein [25, 34–37] using generalized parameterizations or principle components. It is shown in[26, 38, 45] that the redshift dependence of f ( z ) can be ignored up to a ﬁrst approximation,since current CMB data are sensitive to energy injection over a relatively narrow range ofredshift, typically z ∼ − f ( z ) can be replace with a constant f , which wetake as 1 for our analysis. Here we use ’on-the-spot’ approximation, which assumes that theenergy released due to dark matter annihilation is absorbed by IGM locally [24, 40, 41].The terms appearing on the right hand side of Eq. (8) are related to the rate of energyrelease as I Xi = Cχ i [ dE/dt ] n H ( z ) E i (10) I Xα = (1 − C ) χ α [ dE/dt ] n H ( z ) E α . (11)Here E i is the average ionization energy per baryon, E α is the diﬀerence in binding energybetween the 1 s and 2 p energy levels of a hydrogen atom, n H is the number density ofHydrogen Nuclei, and χ i and χ α represent the fraction of energy going ionization and Ly − α photons respectively; which can be expressed in terms of free electron fraction as χ i = χ α =(1 − χ e ) / T b bycontributing one extra term K h as(1 + z ) dT b dz = 8 σ T a R T CMB m e cH ( z ) χ e f He + χ e ( T b − T CMB ) − k B H ( z ) k h f He + χ e + 2 T b . (12)Here the nonstandard term K h arising due to annihilating dark matter is given in terms ofrate of energy release as K h = χ h ( dE/dt ) n H ( z ) , (13)with χ h = (1 + 2 χ e ) / p ann that dependson the properties of dark matter particles as p ann = f h σv i m χ . (14)5he current constraint on p ann with velocity independent h σv i is 1 . × − m s -1 kg -1 C.L from Planck-2018 [27]. In our analysis we will use various electromagnetic formfactors and mass of the dark matter as our model parameters rather than p ann . Hence wewill express energy deposition rate in terms of these parameters for annihilating dark matterwith anapole and dipole moments. The annihilation cross-section for dark matter with anapole moment is given as [12], h σv i χχ → e + e − = 2 g A αm χ v rel . (15)where α = e / (4 π ) and v rel is average relative velocity between the annihilating dark matterparticles in the centre of mass frame. The thermally averaged velocity can be expressed interms of temperature by (cid:0) m χ (cid:1) h v rel i = T . Hence the cross-section (15) can be expressedin terms of temperature as h σv i χχ → e + e − = 4 g A αm χ (cid:18) Tm χ (cid:19) . (16)After decoupling the temperature of the dark matter behaves as T ∝ (1 + z ) . Assumingthe decoupling temperature of the dark matter T d of the order of m χ we get T = T d (1 + z ) (1 + z d ) = T d T T γd (1 + z ) = 10 T m χ (1 + z ) . (17)Here z d and T γd are the redshift of dark matter decoupling and the temperature of photonsat that redshift respectively, which is same as T d . T is the current temperature of CMB.Using Eq. (17) the annihilation cross-section (16) becomes h σv i χχ → e + e − = 40 g A αT (1 + z ) . (18)Hence using Eqs. (18) and (14) we can obtain the expression for p ann as p ann = 40 g A αT m χ (1 + z ) . (19)As mentioned earlier we will choose f ∼ p ann is velocity dependent, the rateof energy release given by Eq. (9) will be dEdt = ρ c c Ω DM g A αT m χ (1 + z ) . (20)Here the redshift dependence of the energy deposition rate is modiﬁed as compared to (9).6 .2. Dark matter with electric dipole moment For DM with electric dipole moment the annihilation cross-section is given by [11], h σv i χχ → e + e − = α D v rel (21)where v rel is the relative velocity of two annihilating WIMPS. For thermal averaged cross-section T = m χ h v rel i /

3. So the annihilation cross-section for dark matter can be expressedin terms of temperature as h σv i χχ → e + e − = α D (cid:18) Tm χ (cid:19) . (22)Assuming T d ∼ m χ and using Eq. (17) for temperature of the dark matter the annihilationcross-section for dark matter with electric dipole moment becomes h σv i χχ → e + e − = 5 α D T m χ (1 + z ) . (23)Hence using (14) we get p ann = 5 α D T m χ (1 + z ) . (24)In this case the energy deposition rate will be dEdt = 12 ρ c c Ω DM α D T m χ (1 + z ) . (25)Here also the redshift dependence is modiﬁed as compared to (9), since the thermally aver-aged cross-section is velocity dependent. For dark matter with magnetic dipole moment the annihilation cross-section is given as[11], h σv i χχ → e + e − = αµ , (26)and hence p ann = αµ m χ . (27)Here the annihilation cross-section does not depend on the velocity of dark matter so theenergy deposition rate will be dEdt = 12 ρ c c Ω DM αµ m χ (1 + z ) , . (28)which has the same redshift dependence as in Eq. (9).7 .4. Rayleigh dark matter For scalar dark matter with Rayleigh coupling (4) the annihilation cross-section is givenby [17], h σv i φφ → γγ = ( g ) m φ Λ , (29)and hence p ann = ( g ) m φ Λ . (30)Here again the annihilation cross-section is independent of the velocity of dark matter, sothe energy deposition rate will be same as (9). dEdt = 12 ρ c c Ω DM ( g ) m φ Λ (1 + z ) . (31)

4. CMB CONSTRAINTS ON VARIOUS MULTIPOLE MOMENTS OF DARKMATTER

As mentioned earlier annihilating dark matter increases the ionization fraction duringrecombination and heats the plasma. Hence the evolution equations of free electron fractionand matter temperature get modiﬁed as given by Eq. (5) and Eq. (12) respectively. Thenon-standard ionization rate I X to compute free electron fraction can be obtained using Eq.(8) along with Eqs. (10) and (11). We use these equations along with energy depositionrates (20), (25), (28) and (31) for dark matter with anapole moment, electric dipole momentand magnetic dipole moment, and Rayleigh coupling to modify RECFAST routine [33] inCAMB [42]. We have also checked our analysis using CosmoRec [43] and HyRec [44, 45] codeinstead of RECFAST, and we found similar results. With this we obtain modiﬁed theoreticalangular power spectra, which can be used to compute the bounds on various electromagneticform factors and mass of the dark matter from Planck-2018 data using COSMOMC [46].The priors for the multipole moments and mass of the dark matter are given in Table I.All these priors are sampled logarithmically to cover a larger range for the new parameters.We also vary the other six parameters of ΛCDM model with priors given in [47]. We haveimposed ﬂat priors for all parameters. Type of dark matter coupling PriorsAnapole 5 . < ln (cid:0) (cid:0) g A /GeV − (cid:1)(cid:1) < − . < log ( m χ /GeV ) < . − . < ln(10 ( D / ( e − cm )) < − . < log ( m χ /GeV ) < . − . < ln(10 ( µ/µ B )) < . − . < log ( m χ /GeV ) < . − . < ln(10 g / (Λ GeV − )) < . − . < log ( m χ /GeV ) < .

8e use the lower bound for the mass of dark matter with anapole moment, and electricand magnetic dipole moment as 1 MeV since the annihilation channel for this case is χ χ → e + e − . However, in case of scalar dark matter with Rayleigh coupling, [17], the dark matterannihilates to photons having energy around 1 eV during recombination, which is used as thelower bound for mass of Rayleigh dark matter. We also use BAO and Pantheon data alongwith Planck-2018 observations for our analysis. We perform MCMC convergence diagnostictests on 4 chains using the Gelman and Rubin ”variance of mean”/”mean of chain variance”R-1 statistics for each parameter.The constraints obtained for anapole moment and mass of the dark matter along withother six parameters of ΛCDM model are shown in Table II. Fig. 1 represents the marginal-ized constraints on anapole moment and mass of the dark matter along with joint 68% CLand 95% CL constraints on both the parameters from Planck-2018 and BAO data. Parameter 68% limits 95% limits 99% limits Ω b h . ± . . +0 . − . . +0 . − . Ω c h . ± . . +0 . − . . +0 . − . τ . ± . . +0 . − . . +0 . − . ln(10 ( g A /GeV − )) < . < . < . log ( m χ /GeV ) − − − − − − − − − ln(10 A s ) . ± .

014 3 . +0 . − . . +0 . − . n s . ± . . +0 . − . . +0 . − . H . ± .

41 67 . +0 . − . . +1 . − . TABLE II: Planck-2018 and BAO constraints on anapole momentum and mass of the dark matterwith other 6 parameters of ΛCDM

10 20 30 40ln(10 (g A /GeV −2 ))0.00.51.0 P r o b a b ili t y −2 −1 0 1log (m χ /GeV)0.00.51.0 P r o b a b ili t y (a)Marginalized constraints

10 20 30 40ln(10 (g A /GeV −2 ))−3−2−1012 l o g ( m χ / G e V ) (b)Joint 68% CL and 95%CL constraints FIG. 1: Constraints for anapole moment and mass of the dark matter using Planck-2018 and BAOdata

9t can be seen from Table II that g A < . × GeV -2 C.L. (32)The constraints obtained using Planck-2018 and BAO data on electric dipole momentand the mass of the dark matter along with the other six parameters of ΛCDM model arelisted in Table. III. Fig. 2 depicts the marginalized constraints on electric dipole momentand mass of the dark matter and with joint 68% CL and 95%CL constraints on both theparameters from Planck-2018 and BAO data.

Parameter 68% limits 95% limits 99% limits Ω b h . ± . . +0 . − . . +0 . − . Ω c h . ± . . +0 . − . . +0 . − . τ . ± . . +0 . − . . +0 . − . ln(10 ( D / ( e − cm )) < . < . < . log ( m χ /GeV ) > − . − − − − − − ln(10 A s ) . ± .

014 3 . +0 . − . . +0 . − . n s . ± . . +0 . − . . +0 . − . H . ± .

41 67 . +0 . − . . +1 . − . TABLE III: Planck-2018 and BAO constraints on electric dipole momentum and mass of the darkmatter with other 6 parameters of ΛCDM (/(e−cm))0.00.51.0 P r o b a b ili t y −2 −1 0 1log (m χ /GeV)0.00.51.0 P r o b a b ili t y (a)Marginalized constraints (/(e−cm))−3−2−1012 l o g ( m χ / G e V ) (b)Joint 68% CL and 95%CL constraints FIG. 2: Constraints for electric dipole moment and mass of the dark matter using Planck-2018and BAO data.

We can see from Table. III that D < . × − e-cm 95% C.L. (33)10able. IV represents the constraints on magnetic dipole moment and mass of the darkmatter obtained from Planck-2018 and BAO data. Here also we have quoted the constraintson other six parameters of ΛCDM. Fig. 3 represents the marginalized constraints on magneticdipole moment and mass of the dark matter along with joint 68% CL and 95%CL constraintson both the parameters.

Parameter 68% limits 95% limits 99% limits Ω b h . ± . . +0 . − . . +0 . − . Ω c h . ± . . +0 . − . . +0 . − . τ . ± . . +0 . − . . +0 . − . ln(10 ( µ/µ B )) − . +4 . − . < . < . log ( m χ /GeV ) − − − − − − .... ln(10 A s ) . ± .

014 3 . +0 . − . . +0 . − . n s . ± . . +0 . − . . +0 . − . H . ± .

41 67 . +0 . − . . +1 . − . TABLE IV: Planck-2018 and BAO constraints on magnetic dipole momentum and mass of Majo-rana fermion dark matter with other 6 parameters of ΛCDM −5 0 5 10ln(10 (μ/μ B ))0.00.51.0 P r o b a b ili t y −2 −1 0 1log (m χ /GeV)0.00.51.0 P r o b a b ili t y (a)Marginalized constraints −10 −5 0 5 10ln(10 (μ/μ B ))−3−2−1012 l o g ( m χ / G e V ) (b)Joint 68% CL and 95%CL constraints FIG. 3: Constraints for magnetic dipole moment and mass of the dark matter using Planck-2018and BAO data.

Again we can read from Table IV that µ < . × − µ B C.L. (34)Similarly Table. IV lists the constraints on Rayleigh coupling and mass of the scalar darkmatter. Fig. 3 represents the marginalized constraints on both of these parameters alongwith joint 68% CL and 95%CL constraints from Planck-2018 and BAO data. Again we havementioned constraints on other six parameters of ΛCDM.11 arameter 68% limits 95% limits 99% limits Ω b h . ± . . +0 . − . . +0 . − . Ω c h . ± . . +0 . − . . +0 . − . τ . ± . . +0 . − . . +0 . − . ln(10 g / (Λ GeV − )) < . < . < . log ( m χ /GeV ) < − .

55 — — ln(10 A s ) . ± .

015 3 . +0 . − . . +0 . − . n s . ± . . +0 . − . . +0 . − . H . ± .

41 67 . +0 . − . . +1 . − . TABLE V: Planck-2018 and BAO constraints on Rayleigh coupling and mass of dark matter withother 6 parameters of ΛCDM g /(Λ GeV −2 ))0.00.51.0 P r o b a b ili t y −10 −8 −6 −4 −2 0log (m χ /GeV)0.00.51.0 P r o b a b ili t y (a)Marginalized constraints −10 0 10 20ln(10 g /(Λ GeV −2 ))−10−50 l o g ( m χ / G e V ) (b)Joint 68% CL and 95%CL constraints FIG. 4: Constraints on Rayleigh coupling and mass of scalar dark matter using Planck-2018 andBAO data

We can see from the Table V that g A Λ < . × − GeV -2 C.L. (35)The upper bounds given by Eqns. (32), (33), (34) and (35) are obtained after margenalizingover all other parameters. 12 . CONCLUSIONS

Electromagnetic form factors are an important class of interactions in the eﬀective the-ories framework of classifying dark matter interactions. Electromagnetic dark matter canbe observed bit only via electron scattering direct detection experiments but can also beconstrained from the CMB. Dirac fermions dark matter with non-zero electric and magneticdipole moments can give the correct relic density Ω m h = 0 .

11 by the χχ ↔ f ¯ f freeze-outprocess if D = 2 . × − e-cm and µ = 8 . × − µ B respectively [11]. Majorana fermionswith anapole moment of mass 10 MeV with anapole moment g A = 0 . − can be darkmatter with the correct freeze-out relic density [12].In this paper we have considered anapole and dipolar dark matter matter with masses m χ > O (MeV). We ﬁnd that dark matter with electromagnetic dipole or anapole formfactors will distort the CMB during recombination era by producing relativistic electron viathe process χχ → e + e − . We ﬁnd that the Planck data gives the bounds on electromagneticform factors D < . × − e-cm and µ < . × − µ B , and g A < . × Gev − .Dark matter with O (MeV) mass in thermal equilibrium with radiation will be ruled outby BBN constraints on N eﬀ . These can only have be created after the BBN era by thefreeze-in mechanism. Freeze-in requires very small couplings and our bounds on electricdipole and anapole moments also rules out the freeze-out mechanism for relic density. Thesemay be produced by the freeze-in mechanism which requires smaller couplings [2, 50].For scalar dark matter there is the dimension-six Rayleigh operator coupling with pho-tons. We put the bound on the Rayleigh coupling as g Λ < . × − GeV -2 ( 95%C.L).Thisbound is valid for dark matter mass as low as O (eV). Such light dark matter can only beproduced by the freeze-in mechanism to evade bounds from BBN.Spectral distortion of the CMB can also arise from radiatively decaying dark matter [51].The bounds derived on the radiative lifetime can be used for deriving bounds on dipolarcouplings of Majorana dark matter which can have non-zero transition electric and magneticmoments dipole.

6. ACKNOWLEDGEMENTS

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