Axial gravitational perturbations of slowly rotating compact objects in general relativity and beyond

We study the axial gravitational perturbations of slowly rotating compact objects which are assumed to be supported by anisotropic fluids. We find that the gravitational perturbations decouple from the matter perturbations for axial sectors. We obtain a master wave equation whose potential is fully determined by the metric functions. This equation makes the calculations of gravitational quasi-normal modes for rotating compact objects extremely easy in specific background configurations.