  The existence of a light mediator is beneficial to some phenomena in
astroparticle physics, such as the core-cusp problem and diversity problem. It
can decouple from Standard Model to avoid direct detection constraints,
generally realized by retard decay of the mediator. Their out-of-equilibrium
decay process changes the dark matter (DM) freeze-out via temperature
discrepancy. This type of hidden sector (HS) typically requires a precision
calculation of the freeze-out process considering HS temperature evolution and
the thermal average of the cross-section. If the mediator is light
sufficiently, we can not ignore the s-wave radiative bound state formation
process from the perspective of CMB ionization and Sommerfeld enhancement. We
put large mass splitting between DM and mediator, different temperature
evolution on the same theoretical footing, discussing the implication for DM
relic density in this HS. We study this model and illustrate its property by
considering the general Higgs-portal dark matter scenario, which includes all
the relevant constraints and signals. It shows that the combination of BBN and
CMB constraint favors the not-too-hot HS, $r_{\mathrm{inf}}<10^2$, for the
positive cubic interaction of mediator scenario. On the other hand, the
negative cubic interaction is ruled out except for our proposed blind spot
scenario.
