  We consider theories in which the generation of neutrino masses is associated
with the breaking of an approximate global lepton number symmetry. In such a
scenario the spectrum of light states includes the Majoron, the pseudo-Nambu
Goldstone boson associated with the breaking of the global symmetry. For a
broad range of parameters, the Majoron decays to neutrinos at late times, after
the cosmic neutrinos have decoupled from the thermal bath, resulting in a
secondary contribution to the cosmic neutrino background. We determine the
current bounds on this scenario, and explore the possibility of directly
detecting this secondary cosmic neutrino background in experiments based on
neutrino capture on nuclei. For Majoron masses in the eV range or below, the
neutrino flux from these decays can be comparable to that from the primary
cosmic neutrino background, making it a promising target for direct detection
experiments. The neutrinos from Majoron decay are redshifted by the cosmic
expansion, and exhibit a characteristic energy spectrum that depends on both
the Majoron mass and its lifetime. For Majoron lifetimes of order the age of
the universe or larger, there is also a monochromatic contribution to the
neutrino flux from Majoron decays in the Milky Way that can be comparable to
the diffuse extragalactic flux. We find that for Majoron masses in the eV
range, direct detection experiments based on neutrino capture on tritium, such
as PTOLEMY, will be sensitive to this scenario with 100 gram-years of data. In
the event of a signal, the galactic and extragalactic components can be
distinguished on the basis of their distinct energy distributions, and also by
using directional information obtained by polarizing the target nuclei.
