Possible golden events for ringdown gravitational waves

There is a forbidden region in the parameter space of quasinormal modes of black holes in general relativity. Using both inspiral and ringdown phases of gravitational waves from binary black holes, we propose two methods to test general relativity. We also evaluate how our methods will work when we apply them to Pop III black-hole binaries with typical masses. Adopting the simple mean of the estimated range of the event rate, we have the expected rate of $500{\mathrm{yr}}^{-1}$. Then, the rates of events with signal-to-noise ratios greater than 20 and greater than 50 are $32{\mathrm{yr}}^{-1}$ and $2{\mathrm{yr}}^{-1}$, respectively. Therefore, there is a good chance to confirm (or refute) the Einstein theory in the strong gravity region by observing the expected quasinormal modes.

Figure 1

Fitting curve based on the sensitivity curve of KAGRA [bKAGRA, VRSE(D) configuration] shown in Ref. [32].

Figure 2

Real ($f_R$) and imaginary ($f_I$) parts of QNM frequencies for the dominant $(\ell =2,m=2)$ least-damped $(n=0)$ mode. The (black) thick line shows the Schwarzschild limit, and the (red) curves are for various mass cases terminated at the spin $\alpha =0.998$ [38]. From the top of the (red) curves, we are considering BH masses, $M/M_{\odot }=30,60,90,120,150$, and 180, respectively. The (red) circles for each line denote the spin dependence $\alpha =0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8$, and 0.9 from the left.

Figure 3

In the ($f_R,f_I$) plane, the left and right panels show the parameter estimation in the cases with $\mathrm{SNR}=20$ and 50 for the typical case (with $M_{\text{rem}}=57.0904M_{\odot }$ and ${\alpha }_{\text{rem}}=0.686710$), respectively. The (black) thick line shows the Schwarzschild limit, which is same as that in Fig. 2, and the ellipses are the contours of $1\sigma ,2\sigma ,3\sigma ,4\sigma$, and $5\sigma$. Here, the time and phase parameters ($t_0,{\varphi }_0$) have been marginalized out.

Figure 4

In the ($f_R,f_I$) plane, this figure shows the parameter estimation in the cases with $\mathrm{SNR}=50$ for a Schwarzschild black hole with $M_{\text{rem}}/M_{\odot }=45$ (right), 60 (center), and 90 (left). The (black) thick line shows the Schwarzschild limit which is same as that in Fig. 2, and the ellipses are the contours of $1\sigma ,2\sigma ,3\sigma ,4\sigma$, and $5\sigma$. Here, the time and phase parameters ($t_0,{\varphi }_0$) have been marginalized out.

Figure 5

The QNM frequency expected in GR ($5\sigma$ level) from the inspiral phase with the total mass $M=60M_{\odot }$ and symmetric mass ratio $\eta =1/4$ [the (red) filled region]. The (black) thick line shows the Schwarzschild limit, which is same as that in Fig. 2.