Probing positron cooling in noble gases via annihilation $\gamma$ spectra

$\gamma$ spectra for positron annihilation in noble-gas atoms are calculated using many-body theory for positron momenta up to the positronium-formation threshold. This data is used, together with time-evolving positron-momentum distributions determined in [arXiv:1706.01434 (2017)], to calculate the time-varying $\gamma$ spectra produced during positron cooling in noble gases. The $\gamma$-spectra and their $\bar{S}$ and $\bar{W}$ 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.