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

We consider dark matter which have non-zero electromagnetic form factors like electric/magnetic dipole moments and anapole moment for fermionic dark matter and Rayleigh form factor for scalar dark matter. We consider dark matter mass $m_\chi > \cal{ O}({\rm MeV})$ and put constraints on their mass and electromagnetic couplings from CMB and LSS observations. Fermionic dark matter with non-zero electromagnetic form factors can annihilate to $e^+ e^-$ and scalar dark matter can annihilate to $2\gamma$ at the time of recombination and distort the CMB. We analyze dark matter with multipole moments with Planck and BAO observations. We find upper bounds on anapole moment $g_{A}<7.163\times 10^{3} \text{GeV}^{-2}$, electric dipole moment ${\cal D}<7.978\times 10^{-9} \text{e-cm}$, magnetic dipole moment ${\mu}<2.959\times 10^{-7} \mu_B$, and the bound on Rayleigh form factor of dark matter is $g_4/\Lambda_4^2<1.085\times 10^{-2}\text{GeV}^{-2}$ with $95\%$C.L.