  Beliaev damping in a superfluid is the decay of a collective excitation into
two lower frequency collective excitations; it represents the only decay mode
for a bosonic collective excitation in a superfluid at T = 0. The standard
treatment for this decay assumes a linear spectrum, which in turn implies that
the final state momenta must be collinear to the initial state. We extend this
treatment, showing that the inclusion of a gradient term in the Hamiltonian
yields a realistic spectrum for the bosonic excitations; we then derive a
formula for the decay rate of such excitations, and show that even moderate
nonlinearities in the spectrum can yield substantial deviations from the
standard result. We apply our result to an attractive Fermi gas in the BCS-BEC
crossover: here the low-energy bosonic collective excitations are density
oscillations driven by the phase of the pairing order field. These collective
excitations, which are gapless modes as a consequence of the Goldstone
mechanism, have a spectrum which is well established both theoretically and
experimentally, and whose linewidth, we show, is determined at low temperatures
by the Beliaev decay mechanism.
