Gravitational production of nearly thermal fermionic Dark Matter
  We consider the cosmological production of fermionic dark matter during
inflation and a post-inflationary radiation dominated era. This fermion
\emph{only interacts gravitationally}, has a mass $m$ much smaller than the
Hubble scale during inflation (but is otherwise arbitrary) and is in its
Bunch-Davies vacuum state during inflation. We focus on superhorizon modes at
the end of inflation, and assume instantaneous reheating. We obtain the full
energy momentum tensor discussing its renormalization, and show that the
contribution from particle production is of the kinetic-fluid form near
matter-radiation equality. We find \emph{exactly} the distribution function of
produced particles
$|B(k)|^2=\frac{1}{2}\Big[1-(1-e^{-\frac{k^2}{2mT_H}})^{1/2}\Big]$ which
exhibits an " emergent temperature" $T_H=H_0\sqrt{\Omega_R}\simeq
10^{-36}(\mathrm{eV})$. The abundance of the produced particles $\Omega_{pp}$
is very similar to that of a non-relativistic degree of freedom thermalized at
temperature $T_H$, $\Omega_{pp} \propto m \, (m\,T_H)^{3/2}\simeq
\big(m/10^8\mathrm{GeV})\big)^{5/2}$ and "cold" equation of state $w(a) \simeq
(T_H/m a^2)$, both dominated by superhorizon modes at the end of inflation. We
discuss subtle aspects of isocurvature perturbations.
