Stochastic conversions of TeV photons into axion-like particles in
  extragalactic magnetic fields
  Very-high energy photons emitted by distant cosmic sources are absorbed on
the extragalactic background light (EBL) during their propagation. This effect
can be characterized in terms of a photon transfer function at Earth. The
presence of extragalactic magnetic fields could also induce conversions between
very high-energy photons and hypothetical axion-like particles (ALPs). The
turbulent structure of the extragalactic magnetic fields would produce a
stochastic behaviour in these conversions, leading to a statistical
distribution of the photon transfer functions for the different realizations of
the random magnetic fields. To characterize this effect, we derive new
equations to calculate the mean and the variance of this distribution. We find
that, in presence of ALP conversions, the photon transfer functions on
different lines of sight could have relevant deviations with respect to the
mean value, producing both an enhancement or a suppression in the observable
photon flux with respect to the expectations with only absorption. As a
consequence, the most striking signature of the mixing with ALPs would be a
reconstructed EBL density from TeV photon observations which appears to vary
over different directions of the sky: consistent with standard expectations in
some regions, but inconsistent in others.
