Non-adiabatic cosmological production of ultra-light Dark Matter
  We study the non-adiabatic cosmological production of ultra light dark matter
(ULDM) under a minimal set of assumptions: a free ultra light real scalar as a
spectator field in its Bunch-Davies vacuum state during inflation and
instantaneous reheating into a radiation dominated era. For (ULDM) fields
minimally coupled to gravity, non-adiabatic particle production yields a
\emph{distribution function} peaked at \emph{low} comoving momentum
$\mathcal{N}_k \propto 1/k^3$. The infrared behavior is a remnant of the
infrared enhancement of light minimally coupled fields during inflation. We
obtain the full energy momentum tensor, show explicity its equivalence with the
fluid-kinetic one in the adiabatic regime, and extract the abundance, equation
of state and free streaming length (cutoff in the matter power spectrum).
Taking the upper bound on the scale of inflation from Planck, the (UDLM)
saturates the dark matter abundance for $m \simeq 1.5\,\times
10^{-5}\mathrm{eV}$ with an equation of state parameter $w \simeq 10^{-14}$ and
a free streaming length $\lambda_{fs} \simeq 70\,\mathrm{pc}$. Thus this
cosmologically produced (ULDM) yields a \emph{cold} dark matter particle. We
argue that the abundance from non-adiabatic production yields a \emph{lower
bound} on generic (ULDM) and axion-like particles that must be included in any
assessment of (ULDM) as a dark matter candidate.
