Self-interacting dark matter to freeze-in via vector portal

It is challenging to resolve the small-scale problem for dark matter being a weakly-interacting massive particle. We attempt to address this issue by proposing a self-interacting freeze-in dark matter via dark photon. In this model, the dark matter obtains the observed relic abundance via Standard Model $\gamma$ and $Z$ boson induced freeze-in processes, whereas the dark matter force mediator has a negligible relic abundance and a lifetime larger than the age of Universe. We place constraints in classical and resonant regime resolving the small-scale problem from CMB, $X/\gamma$-ray, Supernova 1987A and out-of-equilibrium condition. It turns out that the CMB constraint on dark matter annihilations is satisfied despite large Sommerfeld effect taking place, while the other constraints are trivially accommodated due to various millicharge induced suppressions. Finally we briefly discuss future cosmological tests on such freeze-in dark matter model.