  If two annihilation products of dark matter (DM) particles are
non-relativistic and coupled to a light force mediator, their plane wave
functions are modified due to multiple exchanges of the force mediators. This
gives rise to the Sommerfeld effect (SE). We consider the attractive and
repulsive force SE on the relic density in different phases of freeze-in DM. We
find that in the pure freeze-in region, the attractive/repulsive force SE
slightly increases/decreases DM relic density by less than $20\%$ for TeV-scale
DM. In the reannihilation region, if the portal coupling $\kappa$ is
sufficiently large (by comparing the portal reaction rate to the Hubble rate),
DM density will reach its equilibrium, and subsequently freeze out. Compared to
the case without the SE, the presence of the attractive SE leads to an enlarged
cross-section. As a result, a higher equilibrium value of DM density is
reached, and a lower relic density is obtained after the subsequent freeze-out.
However, the repulsive SE has the opposite influence. In the dark sector (DS)
freeze-out region, also known as the middle flat plateau or ``mesa'' in the
phase diagram, the SE has a significant impact on DM relic abundance. In this
region, the attractive SE suppresses DM relic density by simultaneously
enlarging the cross-section of the portal and DS internal interaction. In
contrast, the repulsive SE will have the opposite effect. Finally, in the usual
freeze-out region, DM relic density is suppressed or enhanced by an enlarged or
reduced cross-section of the portal, respectively, due to the presence of the
attractive or repulsive SE. In summary, when considering the constraint of
producing correct DM relic abundance, the inclusion of SE in the portal
reaction or DS internal reaction will modify the model parameters, resulting in
a band-like possible parameter space.
