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Bounds on QCD axion mass and primordial magnetic field from CMB $μ$-distortion

The oscillation of the CMB photons into axions can cause CMB spectral distortion in the presence of large scale magnetic field. With the COBE limit on the $μ$ parameter and a homogeneous magnetic field with strength $B\lesssim 3.2$ nG at the horizon scale, an upper limit on the axion mass is found to be, $m_a\lesssim 4.8\times 10^{-5}$ eV for the KSVZ axion model. On the other hand, using the value of excluded axion mass $m_a\simeq 3.5\times 10^{-6}$ eV from the ADMX experiment together with the COBE bound on $μ$, is found $B\simeq 46$ nG for the KSVZ axion model and $B\simeq 130$ nG for DFSZ axion model, for a homogeneous magnetic field with coherence length at the present epoch $λ_B\simeq 1.3$ Mpc. Forecast on $B$ and $m_a$ for PIXIE/PRISM expected sensitivity on $μ$ are derived. If CMB $μ$ distortion would be detected by the future space missions PIXIE/PRISM and assuming that the strength of the large scale magnetic field is close to its canonical value, $B\simeq 1-3$ nG, axions in the mass range $2\, μ$eV - $3\, μ$eV would be potential candidates of CMB $μ$-distortion.