Axions are well-motivated candidates for dark matter. Recently, much interest has focused on the detection of photons produced by the resonant conversion of axion dark matter in neutron star magnetospheres. Various groups have begun to obtain radio data to search for the signal, however, more work is needed to obtain a robust theory prediction for the corresponding radio lines. In this work we derive detailed properties for the signal, obtaining both the line shape and time-dependence. The principal physical effects are from refraction in the plasma as well as from gravitation which together lead to substantial lensing which varies over the pulse period. The time-dependence from the co-rotation of the plasma with the pulsar distorts the frequencies leading to a Doppler broadened signal whose width varies in time. For our predictions, we trace curvilinear rays to the line of sight using the full set of equations from Hamiltonian optics for a dispersive medium in curved spacetime. Thus, for the first time, we describe the detailed shape of the line signal as well as its time dependence, which is more pronounced compared to earlier results. Our prediction of the features of the signal will be essential for this kind of dark matter search.
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