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

Quasinormal modes (QNMs) or the 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 has been 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, and 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 perturbations 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 for observations and the no-hair conjecture.

Figure 1

Plot of Eq. (25) for a range of BH masses. For small $M$ $(\sim {10}^{-5}{M}_{\odot })$, the local maxima reappears, making it similar to the scattering potentials $V_{S/V}$. For such small masses (which is only possible for primordial BHs), the massive scalar radiation can propagate to $\infty$ for ${\sigma }^2\ge (6\alpha {)}^{-1}$, and can have a larger share of the net emitted gravitational radiation. For comparison, the green curve shows the potential encountered by $\mathrm{\Phi }$ for a $10M_{\odot }$ BH.

Figure 2

The ratio of the potentials is plotted against the dimensional radius $\tilde{r}$ for different values of $M$ and $\ell =2$. The relative intensity reaches a maximum close to the horizon since the extra mode is localized there, and quickly decreases.