Gravitational production of nearly thermal fermionic dark matter

We consider the cosmological production of fermionic dark matter during inflation and a postinflationary radiation dominated era. This fermion 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 exactly the distribution function of produced particles vertical bar B(k)vertical bar(2) = 1/2 [1 - (1 - e (k2/2mTH))(1/2)], which exhibits an emergent temperature T-H = H-0 root Omega(R) similar or equal to 10(-36) (eV). The energy density of produced particles rho(pp) is very similar to that of a nonrelativistic degree of freedom thermalized at temperature T-H, rho(pp) proportional to m(mT(H))(3/2)/a(3) with abundance Omega(pp) similar or equal to (m/10(8) GeV)(5/2) and cold equation of state w(a) similar or equal to (T-H/ma(2)), both dominated by superhorizon modes at the end of inflation. We discuss subtle aspects of isocurvature perturbations.