Multipolar Dark Matter Freeze-out in an Early Matter-Dominated Universe

The relic abundance of thermal dark matter depends not only on its particle interactions but also on the expansion history of the early Universe. We study the freeze-out of fermionic dark matter interacting with the Standard Model through higher-dimensional electromagnetic operators in an early matter-dominated cosmology. In particular, we consider magnetic dipole, electric dipole, anapole, and charge-radius interactions, and compute the couplings required to reproduce the observed dark matter relic abundance in the presence of entropy injection from the decay of a long-lived heavy field. The resulting parameter space is compared with that obtained in the standard radiation-dominated freeze-out scenario and confront it with current constraints from direct-detection experiments and solar neutrino observations. We find that the entropy dilution associated with an early matter-dominated epoch significantly reduces the interaction strength required to obtain the observed relic abundance, thereby rendering viable regions of parameter space that are excluded in the conventional cosmological history. Our results demonstrate that the cosmological history prior to Big Bang nucleosynthesis can have an important impact on the phenomenology and experimental viability of electromagnetic multipole dark matter.