  Dark matter (DM) could be a relic of freeze-in through a light mediator,
where the DM is produced by extremely feeble, IR-dominated processes in the
thermal Standard Model plasma. In the simplest viable models with the DM mass
below the MeV scale, the DM has a small effective electric charge and is born
with a nonthermal phase-space distribution. This DM candidate would cause
observable departures from standard cosmological evolution. In this work, we
combine data from the cosmic microwave background (CMB), Lyman-$\alpha$ forest,
quasar lensing, stellar streams, and Milky Way satellite abundances to set a
lower limit on freeze-in DM masses up to $\sim 20\,$keV, with the exact
constraint depending on whether the DM thermalizes in its own sector. We
perform forecasts for the CMB-S4 experiment, the Hydrogen Epoch of Reionization
Array, and the Vera Rubin Observatory, finding that freeze-in DM masses up to
$\sim 80\,$keV can be explored. These cosmological probes are highly
complementary with proposed direct-detection efforts to search for this DM
candidate.
