  We consider freeze-in production of 7 keV axino dark matter (DM) in the
supersymmetric Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) model in light of the
3.5 keV line excess. The warmness of such 7 keV DM produced from the thermal
bath, in general, appears in tension with Ly-$\alpha$ forest data, although a
direct comparison is not straightforward. This is because the Ly-$\alpha$
forest constraints are usually reported on the mass of the conventional warm
dark matter (WDM), where large entropy production is implicitly assumed to
occur in the thermal bath after WDM particles decouple. The phase space
distribution of freeze-in axino DM varies depending on production processes and
axino DM may alleviate the tension with the tight Ly-$\alpha$ forest
constraints. By solving the Boltzmann equation, we first obtain the resultant
phase space distribution of axinos produced by 2-body decay, 3-body decay, and
2-to-2 scattering, respectively. The reduced collision term and resultant phase
space distribution are useful for studying other freeze-in scenarios as well.
We then calculate the resultant linear matter power spectra for such axino DM
and directly compare them with the linear matter power spectra for the
conventional WDM. In order to demonstrate realistic axino DM production, we
consider benchmark points with the Higgsino next-to-lightest supersymmetric
particle (NLSP) and wino NLSP. In the case of the Higgsino NLSP, the phase
space distribution of axinos is colder than that in the conventional WDM case,
so the most stringent Ly-$\alpha$ forest constraint can be evaded with mild
entropy production from saxion decay inherent in the supersymmetric DFSZ axion
model.
