We derive structure formation limits on dark matter (DM) composed of keV-scaleaxion-like particles (ALPs), produced via freeze-in through the interactions with photonsand Standard Model (SM) fermions. We employ Lyman-alpha (Ly-α) forest data sets aswell as the observed number of Milky Way (MW) subhalos. We compare results obtainedusing Maxwell-Boltzmann and quantum statistics for describing the SM bath. It should beemphasized that the presence of logarithmic divergences complicates the calculation of theproduction rate, which can not be parameterized with a simple power law behaviour. Theobtained results, in combination with X-ray bounds, exclude the possibility for a photophilic“frozen-in” ALP DM with mass below∼19 keV. For the photophobic ALP scenario, inwhich DM couples primarily to SM fermions, the ALP DM distribution function is peaked atsomewhat lower momentum and hence for such realization we find weaker limits on DM mass.Future facilities, such as the upcoming Vera C. Rubin observatory, will provide measurementswith which the current bounds can be significantly improved to∼80 keV.
