  We perform a comprehensive study of models of dark matter (DM) in a Universe
with a non-thermal cosmological history, i.e with a phase of pressure-less
matter domination before the onset of big-bang nucleosynethesis (BBN). Such
cosmological histories are generically predicted by UV completions that contain
gravitationally coupled scalar fields (moduli). We classify the different
production mechanisms for DM in this framework, generalizing previous works by
considering a wide range of DM masses/couplings and allowing for DM to be in
equilibrium with a "dark" sector. We identify four distinct parametric regimes
for the production of relic DM, and derive accurate semi-analytic
approximations for the DM relic abundance. Our results are particularly
relevant for supersymmetric theories, in which the standard non-thermally
produced DM candidates are disfavored by indirect detection constraints. We
also comment on experimental signals in this framework, focusing on novel
effects involving the power spectrum of DM density perturbations. In
particular, we identify a class of models where the spectrum of DM density
perturbations is sensitive to the pressure-less matter dominated era before
BBN, giving rise to interesting astrophysical signatures to be looked for in
the future. A worthwhile future direction would be to study well-motivated
theoretical models within this framework and carry out detailed studies of the
pattern of expected experimental signals.
