  We study the physics potential of the detection of the Cosmic Neutrino
Background via neutrino capture on tritium, taking the proposed PTOLEMY
experiment as a case study. With the projected energy resolution of $\Delta
\sim$ 0.15 eV, the experiment will be sensitive to neutrino masses with
degenerate spectrum, $m_1 \simeq m_2 \simeq m_3 = m_\nu \gtrsim 0.1$ eV. These
neutrinos are non-relativistic today; 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 $2
m_{\nu}$ above the beta decay endpoint. The signal would exceed the background
from beta decay if the energy resolution is $\Delta \lesssim 0.7 m_\nu $.
Interestingly, the total capture rate depends on the origin of the neutrino
mass, being $\Gamma^{\rm D} \simeq 4$ and $\Gamma^{\rm M} \simeq 8$ events per
year (for a 100 g tritium target) for unclustered Dirac and Majorana neutrinos,
respectively. An enhancement of the rate of up to $\mathcal{O}(1)$ is expected
due to gravitational clustering, with the unique potential to probe the local
overdensity of neutrinos. Turning to more exotic neutrino physics, PTOLEMY
could be sensitive to a lepton asymmetry, and reveal the eV-scale sterile
neutrino that is favored by short baseline oscillation searches. The experiment
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.
