Capturing the Cosmic Neutrino Background on Tritium: The Rate and Resolution Budget from First Principles

The cosmic neutrino background (T = 1.95 K, 56 cm^-3 per flavour per helicity state) has never been directly detected. This paper computes the full budget for detection by neutrino capture on tritium (the PTOLEMY route) from first principles, with every number obtained from the deposited numerical integrals: the capture cross section is derived from the measured tritium half-life (4.15 events per 100 g per year for Dirac neutrinos, 8.30 for Majorana, before clustering); the Galactic clustering enhancement is computed from an NFW halo by Liouville mapping (factors 1.04, 1.18 and 2.56 at neutrino masses 0.05, 0.10 and 0.25 eV); and the endpoint background is computed by convolving the beta spectrum with a Gaussian resolution function, giving a resolution budget (FWHM) of 41, 77 and 183 meV at those masses. A 100 g experiment with 50 meV resolution closes for neutrino masses above approximately 0.062 eV and does not close at 0.05 eV. The Python script and its full output are included.