On the applicability of approximations used in calculation of spectrum of Dark Matter particles produced in particle decays

For the Warm Dark Matter (WDM) candidates the momentum distribution of particles becomes important, since it can be probed with observations of Lyman-$\alpha$ forest structures and confronted with coarse grained phase space density in galaxy clusters. We recall the calculation bt Kaplinghat (2005) of the spectrum in case of dark matter non-thermal production in decays of heavy particles emphasizing on the inherent applicability conditions, which are rather restrictive and sometimes ignored in literature. Cold part of the spectrum requires special care when WDM is considered.