Domain models of turbulent magnetic fields and the spectral shape of ensemble averaged axion-photon conversion

Axion-like particles are light pseudoscalars that appear in many extensions of the Standard Model and are candidates for dark matter. Through their mixing with photons, they imprint energy dependent modulations on the spectra of astrophysical sources shining through magnetised media. When a distant source shines through the turbulent magnetic field of an intervening galaxy cluster, these modulations are stochastic. It has recently been shown that averaging over an ensemble of such sources instead produces a smooth absorption-like feature. In this work we derive the analytic form of this ensemble averaged signal, previously known only numerically. We further relate the index $a$ of the magnetic field energy spectrum to the index $\beta$ of the domain size distribution that simulations use to describe the turbulent field. For $1<a<2$ the two indices obey the simple analytic relation $\beta =3-a$. Any distribution with $\beta <1$ instead mimics the spectral index $a=2$, which the sharp domain walls impose. For Kolmogorov turbulence, relevant for galaxy clusters, the relation gives $\beta =4\mathrm{/}3$.