Light Axinos from Freeze-in: production processes, phase space distributions, and Ly-$\alpha$ forest constraints

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.