Medium induced gluon spectrum in dense inhomogeneous matter

We calculate the spectrum of gluons sourced by the branching of an energetic quark in the presence of an inhomogeneous QCD medium, focusing on the soft radiation limit. We take into account multiple soft interactions between the partons and matter, treating the transverse variations of its parameters within a gradient expansion. Thus, we derive the general form of the medium induced spectrum up to the first order in gradients, and consider its simplifying limits. In particular, we show that to the leading order in matter gradients and using the harmonic approximation for the scattering potential, the full gluon spectrum can be written in a compact closed form suitable for numerical evaluation. The final gluon transverse momentum tends to align along the anisotropy direction, resulting in a non-trivial azimuthal pattern in the jet substructure.