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.
