  Many extensions of the Standard Model include axions or axion-like particles
(ALPs). Here we study ALP to photon conversion in the magnetic field of the
Milky Way and starburst galaxies. By modelling the effects of the coherent and
random magnetic fields, the warm ionized medium and the warm neutral medium on
the conversion process, we simulate maps of the conversion probability across
the sky for a range of ALP energies. In particular, we consider a diffuse
cosmic ALP background (CAB) analogous to the CMB, whose existence is suggested
by string models of inflation. ALP-photon conversion of a CAB in the magnetic
fields of galaxy clusters has been proposed as an explanation of the cluster
soft X-ray excess. We therefore study the phenomenology and expected photon
signal of CAB propagation in the Milky Way. We find that, for the CAB
parameters required to explain the cluster soft X-ray excess, the photon flux
from ALP-photon conversion in the Milky Way would be unobservably small. The
ALP-photon conversion probability in galaxy clusters is 3 orders of magnitude
higher than that in the Milky Way. Furthermore, the morphology of the
unresolved cosmic X-ray background is incompatible with a significant component
from ALP-photon conversion. We also consider ALP-photon conversion in starburst
galaxies, which host much higher magnetic fields. By considering the clumpy
structure of the galactic plasma, we find that conversion probabilities
comparable to those in clusters may be possible in starburst galaxies.
