  Future direct observations of the Cosmic Neutrino Background (C$\nu$B) have
the potential to explore a neutrino lifetime, especially in the region of the
age of the universe, $t_0=4.35\times 10^{17}\ {\rm s}$. We forecast constraints
on neutrino decay via capture of the C$\nu$B on tritium, with emphasis on the
PTOLEMY-type experiment. In addition, in some cases of invisible neutrino decay
into lighter neutrinos in the Standard Model and invisible particles, we can
constrain not only the neutrino lifetime but also the masses of the invisible
particles. For this purpose, we also formulate the energy spectra of the
lighter neutrinos produced by 2-body and 3-body decays, and those of the
electrons emitted in the process of the detection of the lighter neutrinos.
