γ spectra for positron annihilation in noble-gas atoms are calculated using many-body theory for positron momenta up to the positronium-formation threshold. These data are used, together with time-evolving positron-momentum distributions determined in the preceding Letter [Phys. Rev. Lett. 119, 203403 (2017)], to calculate the time-varying γ spectra produced during positron cooling in noble gases. The γ spectra and their ¯S and ¯W shape parameters are shown to be sensitive probes of the time evolution of the positron momentum distribution and thus provide a means of studying positron cooling that is complementary to positron lifetime spectroscopy.