The Pauli Limit on Relic Neutrino Clustering: Maximum Phase-Space Density and the Densest Halo the Background Can Form

Fermions cannot be packed arbitrarily tightly, and that quantum limit caps how dense the relic background can ever become in a halo. For a 0.10 eV neutrino in a Milky Way-like velocity dispersion, the Pauli limit permits at most 1.31e+00 particles per cubic centimetre, a factor of 0.0 above the cosmic mean; for 0.30 eV the cap rises to 3.52e+01 per cubic centimetre, and in a cluster dispersion a 0.15 eV neutrino is capped at 5.51e+02 per cubic centimetre. This paper derives the cap from the primordial phase-space distribution, compares it with the gravitational clustering estimates, and shows that for allowed masses gravity, not the Pauli limit, is the binding constraint in galaxies, while the ordering reverses in principle for sufficiently light neutrinos in shallow potentials. Computed from stated assumptions, not measured.