We explore the model-independent constraints from cosmology on a dark-matter particle with no prominent standard model interactions that interacts and thermalizes with other particles in a hidden sector. Without specifying detailed hidden-sector particle physics, we characterize the relevant physics by the annihilation cross section, mass, and temperature ratio of the hidden to visible sectors. While encompassing the standard cold WIMP scenario, we do not require the freeze-out process to be nonrelativistic. Rather, freeze-out may also occur when dark matter particles are semirelativistic or relativistic. We solve the Boltzmann equation to find the conditions that hidden-sector dark matter accounts for the observed dark-matter density, satisfies the Tremaine-Gunn bound on dark-matter phase space density, and has a free-streaming length consistent with cosmological constraints on the matter power spectrum. We show that for masses <1.5 keV no region of parameter space satisfies all these constraints. This is a gravitationally-mediated lower bound on the dark-matter mass for any model in which the primary component of dark matter once had efficient interactions -- even if it has never been in equilibrium with the standard model.
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