  We propose signals in the cosmic microwave background to probe the type and
spectrum of neutrino masses. In theories that have spontaneous breaking of
approximate lepton flavor symmetries at or below the weak scale, light
pseudo-Goldstone bosons recouple to the cosmic neutrinos after nucleosynthesis
and affect the acoustic oscillations of the electron-photon fluid during the eV
era. Deviations from the Standard Model are predicted for both the total energy
density in radiation during this epoch, \Delta N_nu, and for the multipole of
the n'th CMB peak at large n, \Delta l_n. The latter signal is difficult to
reproduce other than by scattering of the known neutrinos, and is therefore an
ideal test of our class of theories. In many models, the large shift, \Delta
l_n \approx 8 n_S, depends on the number of neutrino species that scatter via
the pseudo-Goldstone boson interaction. This interaction is proportional to the
neutrino masses, so that the signal reflects the neutrino spectrum. The
prediction for \Delta N_nu is highly model dependent, but can be accurately
computed within any given model. It is very sensitive to the number of
pseudo-Goldstone bosons, and therefore to the underlying symmetries of the
leptons, and is typically in the region of 0.03 < \Delta N_nu < 1. This signal
is significantly larger for Majorana neutrinos than for Dirac neutrinos, and,
like the scattering signal, varies as the spectrum of neutrinos is changed from
hierarchical to inverse hierarchical to degenerate.
