If the dark matter is produced in the early universe prior to Big Bang
nucleosynthesis, a modified cosmological history can drastically affect the
abundance of relic dark matter particles. Here, we assume that an additional
species to radiation dominates at early times, causing the expansion rate at a
given temperature to be larger than in the standard radiation-dominated case.
We demonstrate that, if this is the case, dark matter production via freeze-in
(a scenario when dark matter interacts very weakly, and is dumped in the early
universe out of equilibrium by decay or scattering processes involving
particles in the thermal bath) is dramatically suppressed. We illustrate and
quantitatively and analytically study this phenomenon for three different
paradigmatic classes of freeze-in scenarios. For the frozen-in dark matter
abundance to be as large as observations, couplings between the dark matter and
visible-sector particles must be enhanced by several orders of magnitude. This
sheds some optimistic prospects for the otherwise dire experimental and
observational outlook of detecting dark matter produced by freeze-in.