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We study the free-streaming effect in a light freeze-in dark matter model.Naturally in the dark sector one can find dark matter related coupling, andsuch coupling may induce dark matter self-scattering. In case that suchscattering is subdominant, the dark matter partition function is not thermalbut determined by the freeze-in process, yet its high momentum side isgenerally also Boltzmann suppressed. We show that the matter power spectrum isvery similar to a warm dark matter one in shape. When matched to the currentWDM bound, a $24$~keV freeze-in dark matter is ruled out at $2\sigma$confidence level. In case that the dark matter self-scattering is strong anddecouples at a very late time, by a new numerical calculation we show that theearly stage Brownian motion indeed protects the power spectrum againstfree-streaming suppression. However, such an effect cannot be characterized bya free-streaming length alone; we find that the self-scattering decoupling timeis another necessary parameter. The currently interested dark matterself-interaction cross section $\sim\text{cm}^2/\text{g}$ is just marginal forsuch protection to be effective.

Matter Power Spectrum of Light Freeze-in Dark Matter: With or without Self-Interaction