Cosmological neutrino entanglement and quantum pressure

Context. The widespread view that cosmological neutrinos, even if massive, are well described since the decoupling redshift z ≈ 1010 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 Schrödinger 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 − 10. 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.