On the origin of entropy of gravitationally produced dark matter: the
  entanglement entropy
  We study the emergence of entropy in gravitational production of dark matter
particles, ultra light scalars minimally coupled to gravity and heavier
fermions, from inflation to radiation domination (RD). Initial conditions
correspond to dark matter fields in their Bunch-Davies vacua during inflation.
The "out" states are correlated particle-antiparticle pairs, and the
distribution function is found in both cases. In the adiabatic regime the
density matrix features rapid decoherence by dephasing from interference
effects in the basis of "out" particle states, effectively reducing it to a
diagonal form with a concomitant von Neumann entropy. We show that it is
exactly the entanglement entropy obtained by tracing over one member of the
correlated pairs. Remarkably, for both statistics the entanglement entropy is
similar to the quantum kinetic entropy in terms of the distribution function
with noteworthy differences stemming from pair correlations. The entropy and
the kinetic fluid form of the energy momentum tensor all originate from
decoherence of the density matrix. For ultra light scalar dark matter, the
distribution function peaks at low momentum $\propto 1/k^3$ and the specific
entropy is $\ll 1$. This is a hallmark of a \emph{condensed phase} but with
vanishing field expectation value. For fermionic dark matter the distribution
function is nearly thermal and the specific entropy is $\mathcal{O}(1)$ typical
of a thermal species. We argue that the functional form of the entanglement
entropy is quite general and applies to alternative production mechanisms such
as parametric amplification during reheating.
