Deflection of light by a rotating black hole surrounded by "quintessence"

We present a detailed analysis of a rotating black hole surrounded by "quintessence." This solution represents a fluid with a constant equation of state, w, which can for example describe an effective warm dark matter fluid around a black hole. We clarify the conditions for the existence of such a solution and study its structure by analyzing the existence of horizons as well as the extremal case. We show that the deflection angle produced by the black hole depends on the parameters (c, w) which need to obey the condition cw < 0 because of the weak energy condition, where c is an additional parameter describing the hair of the black hole. In this context, we found that for w similar or equal to 0.1 (consistent with warm dark matter) and c < 0, the deviation angle is larger than that in the Kerr space-time for direct and retrograde orbits. We also derive an exact solution in the case of w = -1/3.