Velocity-dependent self-interacting dark matter from thermal freeze-out and tests in direct detections

A small fraction of millicharged dark matter (DM) is considered in the literature to give an interpretation about the enhanced 21-cm absorption at the cosmic dawn. Here we focus on the case that the main component of DM is self-interacting dark matter (SIDM), motivated by the small scale problems. For self interactions of SIDM being compatible from dwarf to cluster scales, velocity-dependent self interactions mediated by a light scalar $\phi$ is considered. To fermionic SIDM $\Psi$, the main annihilation mode $\Psi \bar{\Psi} \to \phi \phi$ is a $p -$wave process. The thermal transition of SIDM $\rightleftarrows \phi \rightleftarrows$ standard model (SM) particles in the early universe sets a lower bound on couplings of $\phi$ to SM particles, which has been excluded by DM direct detections, and here we consider SIDM in the thermal equilibrium via millicharged DM. For $m_\phi >$ twice millicharged DM mass, $\phi$ could decay quickly and avoid excess energy injection to the big bang nucleosynthesis. Thus, the $\phi -$SM particle couplings could be very tiny and evade DM direct detections. The picture of weakly interacting massive particle (WIMP)-nucleus scattering with contact interactions fails for SIDM-nucleus scattering with a light mediator, and a method is explored in this paper, with which a WIMP search result can be converted into the hunt for SIDM in direct detections.