The detection of quasinormal mode with $a/M \sim 0.95$ would prove a sphere $99\%$ soaking in the ergoregion of the Kerr space-time

Recent numerical relativity simulations of mergers of binary black holes suggest that the maximum final value of $a/M$ is $\sim 0.95$ for the coalescence of two equal mass black holes with aligned spins of the same magnitude $a/M=0.994$ which is close to the upper limit $a/M=0.998$ of accretion spin-up shown by Thorne. Using the WKB method, we suggest that if quasinormal modes with $a/M \sim 0.95$ are detected by the second generation gravitational wave detectors, we could confirm the strong gravity space-time based on Einstein's general relativity up to $1.33M$ which is only $\sim 1.014$ times the event horizon radius and within the ergoregion. One more message about black hole geometry is expected here. If the quasinormal mode is different from that of general relativity, we need to find the true theory of gravity which deviates from general relativity only near the black hole horizon.


Introduction
The Kerr space-time [1] which describes a rotating black hole (BH) is very interesting not only in astrophysics but also in mathematics and physics. In particular, the existence of the ergoregion induces various phenomena specific to strongly curved spacetime, such as the Penrose process [2], by which we can extract the rotational energy of BH.
To confirm the space-time described by the Kerr BH using the second generation gravitational wave detectors such as Advanced LIGO (aLIGO) [3], Advanced Virgo (AdV) [4] and KAGRA [5,6], we discuss gravitational waves of the quasinormal modes (QNMs) which are the unique signature of the gravitational wave emission from the BH. We expect that the gravitational waves are emitted when a BH is formed after the merger of compact objects, and for example, the possible detection rate of BH-BH mergers has been discussed in our previous paper [7].
It would be interesting here to observe that there may be a restriction on the spin of BHs astrophysically. From the mass formula of BH [8], we have the gravitational mass of M as where q = a/M with the BH's specific angular momentum a, and M irr is the irreducible mass of the Kerr BH which is related to the area of the event horizon A as A = 16πM 2 irr . For a single BH, Thorne [9] showed that the maximum value of q max is ∼ 0.998 since the radiation emitted by the accretion disk carries the angular momentum to prevent the BH from reaching the extremal limit, q = 1. Now, let us consider the merger of two equal mass Kerr BHs of mass M with the maximum value of aligned spins q = q max , which results in a single BH with the final mass M f and q = q f . Since the area of the horizon should increase [10] (see also Ref. [11]), we have This means that if M f ≥ 1.46M , q f = 1 is possible in contrast to the accretion spin-up model by Thorne [9]. However, even in a recent numerical relativity simulation of a binary BH merger with equal mass M 1 = M 2 = M and aligned equal spins with q 1 = q 2 = 0.994 by Scheel et al. [12], q f is ∼ 0.95. Therefore, we restrict our study up to q f = 0.97 here.
In our previous paper [7], we have presented a method to claim how close to the event horizon of a BH we actually see by gravitational wave detection of the QNMs. In that paper, our focus was not in determining the QNM frequencies accurately at all. Instead, we used the known accurate numerical results of the complex QNM frequencies and the separation constant λ in the Teukolsky equation [13], to suggest which part of strong gravity space-time is confirmed by the detection of QNMs. In the present paper, using the same approach, we discuss further whether we can reach the confirmation of the space-time region within the ergoregion by the QNM gravitational wave detection.
This paper is organized as follows. In § 2, we will argue our method briefly. The results are given is § 3 and § 4 is devoted to discussions. We use the geometric unit system, where G = c = 1 in this paper.

Approach
In the Boyer-Lindquist coordinates, the Kerr metric is given by where M and a are the mass and the spin parameter, respectively, Σ = r 2 + a 2 cos 2 θ and ∆ = r 2 − 2M r + a 2 . We summarize here three characteristic radii in this Kerr space-time. The event horizon is located at and the inner light ring radius [14] is at which is evaluated in the equatorial (θ = π/2) plane. In Ref. [15] (and references therein), the relation between this light ring orbit and the QNM frequencies in the high frequency 2/8 regime has been discussed. Also, the ergosurface is given by and we denote the equatorial radius of the ergoregion as In the previous paper [7], we have extended the physical picture of QNMs by Schutz and Will [16] for the Schwarzschild space-time via the WKB method to the Kerr space-time. Given the radial wave equation, where dr * /dr = (r 2 + a 2 )/∆ and V D is a potential which is obtained from the potential of the Teukolsky radial equation [13]. Here, we approximate the potential by the expansion near the radius at its peak location r * 0 as Here, r * 0 is related to the peak location in the Boyer-Lindquist radial coordinate as where . In practice, we determine r * 0 by evaluating the peak location of the absolute value of the potential in Eq. (8). In the appendix of Ref. [7], we have also discussed the location of r * which satisfies dV /dr * = 0 in the complex radius plane, and then read off the effective peak radius from the real part of the complex radius, to find no significant difference between two radii.
Then, the QNM frequencies are derived as in the leading order WKB analysis.
Here, in Ref. [7], we used the WKB approximation to determine QNMs by using the spatial positions of the maximum absolute values of the Sasaki-Nakamura potential V SN [17][18][19] and the Detweiler potential V D [20] up to q = 0.8 since the remnant spin q f ∼ 0.7 is expected from the results of numerical relativity for the merger of equal mass BHs with q 1 = q 2 = 0 [21][22][23]. In this paper, we consider up to q f = 0.97 since a recent result of numerical relativity for q 1 = q 2 ∼ q max [12] yields q f ∼ 0.95. Also, in this paper, we treat the Detweiler potential [20] (the (−+) case in Ref. [7]) as the potential V D . This is because the (−+) potential has less wavy shape and looks most suitable compared with the other cases for the WKB analysis (see Figs. 1 and 5 of Ref. [7]). We do not repeat how to derive V D since the details are written in Sec. 4 of Ref. [7].
In the following, we focus only on the (ℓ = 2, m = 2) mode. This is because the (ℓ = 2, m = 2) QNM is dominant in numerical relativity simulations of binary BH mergers (see e.g., Ref. [24]) even in the case of the remnant spin q f ∼ 0.95 [12] (we can check the behavior by using the waveforms in "SXS Gravitational Waveform Database" [25]).

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3. Results First, we present the behavior of the real part of the potential Re(V D ), the imaginary part Im(V D ) and the absolute value |V D | for various non-dimensional spin parameters in Figs. 1 and 2. The three panels in Fig. 1 are for q = 0.7 (left), 0.8 (center) and 0.9 (right). We find that the contribution of the imaginary part is small even for q = 0.9. On the other hand, the three panels in Fig. 2 shows the potential for q = 0.93 (left), 0.95 (center) and 0.97 (right). Again, the contribution of the imaginary part is small, but we see another peak around r * /M ∼ 5 in the q = 0.97 case (in practice, we also see another peak in the q = 0.95 case outside the figure). This peak grows for larger q and the height of the peak becomes dominant in the q = 0.98 case. Therefore, we restrict our analysis up to q = 0.97 which is appropriate since the recent numerical relativity results suggest q f 0.95 [12].    In Table 1, we summarize the event horizon radius r + , the peak location (r peak ) of the absolute value of the potential V D , and the location which satisfies dV D /dr * = 0 in the complex radius plane, r dVD/dr * =0 . The differences between the real part of r dVD/dr * =0 and r peak are small, and also the imaginary part of dV D /dr * = 0 is small. In the same table, we also show the WKB result of Im(ω)M and the solid angle of a sphere of r peak soaking in the 4/8 ergoregion (4πC) estimated as where θ m is calculated by r peak = M (1 + 1 − q 2 cos 2 θ m ). The timelike Killing vector field of the Kerr space-time becomes spacelike in the ergoregion, and the region is coordinate invariant. Here, we have introduced this C as an estimator which is less dependent on the coordinates, while the radial coordinate is variant. Table 1 The event horizon radius, the peak location of the absolute value of the potential V D , and the location of r which satisfies dV D /dr * = 0 in the complex radius plane. We also show the WKB result of Im(ω)M . The solid angle of a sphere of r peak soaking in the ergoregion (4πC) is estimated by Eq. (12). The peak location (r peak ) of the absolute value of the potential V D in Table 1 is plotted in Fig. 3. In this figure, we also show the event horizon radius r + , the inner light ring radius r lr , and the equatorial radius of the ergoregion r ergo .
The real and imaginary parts of the n = 0 QNM frequencies are shown in the left panel of Fig. 4, respectively. Here, we present the WKB result and the exact frequencies given in Ref. [26]. We find that the errors are quite small in this spin range (see the right panel of Fig. 4).

Discussions
In our previous paper [7], we assumed that the remnant BH spin q f = 0.7 to compute the detection rate. When we treat BHs with q f = 0.95, two effects from the QNM frequency arise, that is, the (real part of) frequency of QNM increases while the damping rate decreases. Therefore, the recalculation is needed (note that the QNM amplitude (excitation) is also an important input). However, the former effect will decrease the event rate while the latter effect increases the event rate. Therefore, the event rate will be more or less similar. This means that the merging rate of Pop III ∼ 30M ⊙ -30M ⊙ BH binaries with the signal-to-noise ratio > 35 needed for the determination of QNMs [28] is roughly 0.17-7.2 events yr −1 (SFR p /(10 −2.5 M ⊙ yr −1 Mpc −3 )) · ([f b /(1 + f b )]/0.33) where SFR p and f b are the peak value of the Pop III star formation rate and the fraction of binaries, respectively [29,30]. 5 Fig. 3 The location of the maximum of the absolute value of the potential |V D | with (ℓ = 2, m = 2). We also show the event horizon r + /M = 1 + 1 − q 2 , the inner light ring radius r lr /M = 2(1 + cos((2/3) cos −1 (−q))), and the equatorial radius of the ergoregion r ergo /M = 2 evaluated for various spin parameters q = a/M . δ R δ I Fig. 4 (Left) The real and imaginary parts of the fundamental (n = 0) QNM frequencies with V D (ℓ = 2, m = 2) evaluated for various spin parameters q = a/M . The exact frequencies Re(ω) and Im(ω) are from Ref. [26,27]. (Right) Absolute value of relative errors for the real and imaginary part of the QNM frequencies with V D (ℓ = 2, m = 2), δ R = |(WKB Re(ω))/Re(ω) − 1| and δ I = |(WKB Im(ω))/Im(ω) − 1| between the exact value and that of the WKB approximation.
As seen in Fig. 3, the location of the maximum of the absolute value of the potential is within the ergoregion for q 0.7. It is noted that in Table 1 in the range between q = 0.7 and 0.97 (see Fig. 5). Related to the existence of the event horizon, it is important to know how far we can confirm the ergoregion by gravitational 6/8 wave detection of the QNMs. The above relations give us the direct relation between C and the imaginary part of the observed QNM frequency.
Here, according to Hod [31], the imaginary part of the QNM frequency for near extremal Kerr BHs is which is shown as the blue dashed curve in the left panel of Fig. 5. The difference between Eqs. (13) and (14) can be considered as the next order effect of the near-extremal approximation. We will clarify the physical meaning of Eqs. (13) in a future work. Finally, it is necessary to investigate QNMs in the case of q > 0.97 that requires some high accuracy study (see e.g., Ref. [32]). Also, there are many excited states [33] which contribute to the QNM gravitational waves. The detailed study is needed for a fully-integrated understanding of QNMs.
In conclusion, we have covered the QNM analysis of the Kerr BH up to q ∼ 0.95 in this paper. This spin parameter q ∼ 0.95 is expected by state-of-the-art numerical relativity simulations of mergers of a equal-mass, highly spinning BH binary with aligned spins of the same magnitude q = 0.994. The detection of the QNM with a/M ∼ 0.95 by the second generation gravitational wave detectors would prove a sphere 99% soaking in the ergoregion of the Kerr BH, and such a space-time region will give us the opportunity to test Einstein's general relativity in strong gravity.