How warm are non-thermal relics? Lyman-$\alpha$ bounds on
  out-of-equilibrium dark matter
  We investigate the power spectrum of Non-Cold Dark Matter (NCDM) produced in
a state out of thermal equilibrium. We consider dark matter production from the
decay of scalar condensates (inflaton, moduli), the decay of thermalized and
non-thermalized particles, and from thermal and non-thermal freeze-in. For each
case, we compute the NCDM phase space distribution and the linear matter power
spectrum, which features a cutoff analogous to that for Warm Dark Matter (WDM).
This scale is solely determined by the equation of state of NCDM. We propose a
mapping procedure that translates the WDM Lyman-$\alpha$ mass bound to NCDM
scenarios. This procedure does not require expensive ad hoc numerical
computations of the non-linear matter power spectrum. By applying it, we obtain
bounds on several NCDM possibilities, ranging from $m_{\rm DM}\gtrsim {\rm
EeV}$ for DM production from inflaton decay with a low reheating temperature,
to sub-keV values for non-thermal freeze-in. We discuss the phenomenological
implications of these results for specific examples which include
strongly-stabilized and non-stabilized supersymmetric moduli, gravitino
production from inflaton decay, $Z'$ and spin-2 mediated freeze-in, and
non-supersymmetric spin-3/2 DM.
