Cosmological Neutrino Entanglement and Quantum Pressure

Context: The widespread view that cosmological neutrinos, even if massive, are well described since the decoupling redshift z~10^10 down to the present epoch by an almost perfectly collisionless fluid of classical point particles is re-examined. Aims: In view of the likely sub-eV rest mass of neutrinos, the main effects due to their fermionic nature are studied. Methods: By numerical means we calculate the accurate entropy, fugacity and pressure of cosmological neutrinos in the Universe expansion. By solving the Schroedinger equation we derive how and how fast semi-degenerate identical free fermions become entangled. Results: We find that for sub-eV neutrinos the exchange degeneracy has significantly increased during the relativistic to non-relativistic transition epoch at z~10^4-10^5. At all times neutrinos become entangled in less than 10^-6 s, much faster than any plausible decoherence time. The total pressure is increased by quantum effect from 5% at high redshifts to 68% at low redshifts with respect to a collisionless classical fluid. Conclusions: The quantum overpressure has no dynamical consequences in the homogeneous regime at high redshifts, but must be significant for neutrino clustering during the non-linear structure formation epoch at low redshifts.