Quasinormal modes as a distinguisher between general relativity and f(R) gravity

Quasi-Normal Modes (QNM) or ringdown phase of gravitational waves provide critical information about the structure of compact objects like Black Holes. Thus, QNMs can be a tool to test General Relativity (GR) and possible deviations from it. In the case of GR, it is known for a long time that a relation between two types of Black Hole perturbations: scalar (Zerilli) and vector (Regge-Wheeler), leads to an equal share of emitted gravitational energy. With the direct detection of Gravitational waves, it is now natural to ask: whether the same relation (between scalar and vector perturbations) holds for modified gravity theories? If not, whether one can use this as a way to probe deviations from General Relativity. As a first step, we show explicitly that the above relation between Regge-Wheeler and Zerilli breaks down for a general f (R) model, and hence the two perturbations do not share equal amounts of emitted gravitational energy. We discuss the implication of this imbalance on observations and the no-hair conjecture.