I will talk about the physics potential of the detection of the Cosmic Neutrino Background "directly" via neutrino capture on tritium, taking the proposed PTOLEMY experiment as a case study. With the projected energy resolution of 0.15 eV, the experiment will be sensitive to neutrino masses with degenerate spectrum, m_\nu >0.1 eV. These neutrinos are non-relativistic today, hence, detecting them would be a unique opportunity to probe this unexplored kinematical regime. The signature of neutrino capture is a peak in the electron spectrum that is displaced by 2m_\nu above the beta decay endpoint. The signal would exceed the background from beta decay if the energy resolution is better than 0.7 m_\nu. I will show that the total capture rate depends on the origin of the neutrino mass, namely, it is about 4 events per year for Dirac neutrinos, whereas it is 8 events per year for Majorana neutrinos for a 100 g tritium target. An enhancement of the rate of up to O(1) is expected due to gravitational clustering, with the unique potential to probe the local overdensity of neutrinos. Finally, I will briefly talk about what else can be learned about the cosmic neutrino background or about neutrino physics from an experiment like PTOLEMY. The experiment could be sensitive to a lepton asymmetry, and reveal the eV-scale sterile neutrino that is favored by short baseline oscillation searches. It would also be sensitive to a neutrino lifetime on the order of the age of the universe and break the degeneracy between neutrino mass and lifetime which affects existing bounds.
