Simple estimate of BBN sensitivity to light freeze-in dark matter

We provide a simple analysis of the big-bang nucleosynthesis (BBN) sensitivity to the light dark matter (DM) generated by the thermal freeze-in mechanism. It is shown that the ratio of the effective neutrino number shift N_ over the DM relic density h^2, denoted by R_N_/, cancels the decaying particle mass and the feeble coupling, rendering therefore a simple visualization of N_ at the BBN epoch in terms of the DM mass. This property drives one to conclude that the shift with a sensitivity of N_ (0. 1) cannot originate from a single warm DM under the Lyman- forest constraints. For the cold-plus-warm DM scenarios, where the Lyman- constraints are diluted, the ratio R_ can be potentially used to test the thermal freeze-in mechanism in generating a small warm component of DM and a possible excess at the level of N_ (0. 01).