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.