undistinguishable).

Focusing now on the benchmark points $\omega=1$ and $100\,\text{keV}$, $m_a=10^{-8}\,\text{keV}$, $g/e=5\times10^{-17}\,\text{keV}^{-1}$, $r_0=10\,\text{km}$, $B_0=20\times10^{14}\,\text{G}$ and $\theta=\pi/2$, which lie in the soft and hard X-ray range with appropriate magnetar parameters for SGR 1806-20 and are not excluded by CAST, different dependences of $R(\chi_0=0)$ around the benchmark points are shown in Fig. \[FigBP\].

As explained in [@Fortin:2018ehg], the fact that the conversion probability peaks in the X-ray range (as shown in the first panel of Fig. \[FigBP\]) has interesting consequences since the ALP nucleon-nucleon bremsstrahlung emission spectrum peaks in the hard X-ray range for our benchmark model (for a degenerate medium relevant to magnetars). The second panel shows $R$ as a function of the ALP-photon coupling constant $g$. It is shown that the conversion probability drops dramatically for smaller ALP-photon coupling constant, as expected from physical arguments. Indeed, for small ALP-photon coupling constant $g$, there is effectively no oscillations and the conversion probability decreases.

Furthermore, contrary to some of the benchmark points used in [@Fortin:2018ehg], the conversion probability for the CAST-friendly benchmark points used here can be obtained by analogy to time-dependent perturbation theory in quantum mechanics, leading to the approximation [@Raffelt:1987im] \[EqnPapprox\] These equations are correct as long as the numerical 