Possible confirmation of the existence of ergoregion by the Kerr quasinormal mode in gravitational waves from Pop III massive black hole binary

The existence of the ergoregion of the Kerr space-time has not been confirmed observationally yet. We show that the confirmation would be possible by observing the quasinormal mode in gravitational waves. As an example, using the recent population synthesis results of Pop III binary black holes, we find that the peak of the final merger mass ($M_f$) is about $50~\rm M_{\odot}$, while the fraction of the final spin $q_f = a_f/M_f>0.7$ needed for the confirmation of a part of ergoregion is $\sim 77\%$. To confirm the frequency of the quasinormal mode, ${\rm SNR}>35$ is needed. The standard model of Pop III population synthesis tells us that the event rate for the confirmation of more than $50\%$ of the ergoregion by the second generation gravitational wave detectors is $\sim 2.3$ ${\rm events\ yr^{-1}\ (SFR_p/(10^{-2.5}\ M_\odot yr^{-1}\ Mpc^{-3}))} \cdot (\rm [f_b/(1+f_b)]/0.33)$ where ${\rm SFR_p}$ and ${\rm f_b}$ are the peak value of the Pop III star formation rate and the fraction of binaries, respectively.


Introduction
The Kerr space-time [1] is the unique one in two senses. Firstly it is the unique stationary solution [2][3][4] of the Einstein equation in the vacuum under the cosmic censorship [5], which demands the singularity should be covered by the event horizon. Secondly, it has the ergoregion where the timelike Killing vector turns out to be spacelike. This causes various interesting mechanisms to extract the rotational energy of the Kerr black hole (BH) such as Penrose process [5] and the Blanford-Znajek process [6]. Although there are so many papers using these two mechanisms in the fields of physics and astrophysics, so far the existence of the ergoregion of the Kerr BH has not been confirmed observationally yet. In this paper, we suggest a possible method to confirm the existence of at least a part of the ergoregion.
The Kerr space-time in the geometric unit of G = c = 1 is specified by its gravitational mass M and the specific angular momentum a. We use the non-dimensional spin parameter q = a/M instead of a hereafter. There are two important quantities. The first one is the outer event horizon radius r + defined by The second one is the location of the outer boundary of the ergoregion (r ergo (θ)) defined by r ergo (θ) = M 1 + 1 − q 2 cos 2 θ , which is called as the ergosphere. Note that r ergo (0) = r + and r ergo (π/2) = 2M . The quasinormal mode (QNM) is the free oscillation of the Kerr BH after the merger of BH binaries. The complex QNM frequencies are determined by using the Leaver's method [7] accurately. A recent numerical relativity simulation of the BH binary with the initial equal mass and spins of q 1 = q 2 = 0.994 results in the final spin q f ∼ 0.95 [8]. As for the physical meaning of QNMs, Schutz and Will [9] used the WKB method for the q f = 0 case, that is, the Schwarzschild space-time, and they showed that the real and imaginary parts of the QNMs are determined by the peak value and the second derivative , respectively, of the Regge-Wheeler potential [10], which determines the behavior of gravitational perturbations in the Schwarzschild space-time. The location of the peak for the dominant ℓ = 2 mode is at r max = 3.28M , and the errors due to the WKB approximation are about 7% and 0.7% for the real and imaginary parts of the fundamental (n = 0) QNM frequency, respectively. This suggests that the complex frequency of the QNM is determined by the space-time around r max . Conversely, if the ℓ = 2 QNM which is the dominant mode, is confirmed by the second generation gravitational wave detectors, such as Advanced LIGO (aLIGO) [11], Advanced Virgo (AdV) [12], and KAGRA [13,14], we can say that the strong space-time around r = 3.28M is confirmed as predicted by Einstein's general relativity. The reason for the word "around" comes from the fact that the imaginary part of the QNM frequency is determined by the second derivative of the Regge-Wheeler potential.
In the paper submitted to PTEP [15] by Nakano, Nakamura and Tanaka , they showed that the similar physical picture to that presented by Schutz and Will can be obtained by using the Detweiler potential [16] of gravitational perturbations [17] in the Kerr space-time. 1 The maximum errors of the real and imaginary parts of the QNM frequency with (ℓ = m = 2), which is the dominant mode shown by the numerical relativity simulations [20], are 1.5% and 2% in the range of 0.7 < q f < 0.98, respectively. They also obtained that the QNM for q f > 0.7 reflects the Kerr space-time within the ergoregion because r max < 2M . Since the ergoregion radius depends on θ, we can define the covered solid angle 4πC for each r max < 2M by C = cos θ m with θ m defined by r max = M (1 + 1 − q 2 cos 2 θ m ). It is found that an empirical relation between (1 − C) and (1 − q) for 0.7 < q < 0.98 given by exists. The correlation coefficient of this empiriacl relation is 0.989 with the chance probability of 5.9 × 10 −8 . The purpose of this paper is to apply Eq. (3) to the recent population synthesis results of Population III (Pop III) massive BH binaries to know the event rate of detection of the QNM gravitational waves and the typical value of C as an example.
This paper is organized as follows. In §2, we briefly argue the recent population synthesis results of Pop III massive BH binaries [21,22]. The reader who is not familiar with the population synthesis, may skip this section. In §3, we discuss methods to obtain the final BH's mass M f and spin q f with their distribution functions and the detection rate as a function of C. Finally, §4 is devoted the discussions.

2.
Pop III binary calculation PopIII star is the first star in our universe which does not have the metal with atomic number larger than the carbon. To study Pop III binary evolutions, they used a Pop III binary population synthesis code [21,22] which is upgraded from Hurley's BSE code [23,24] for the case of Pop I stars to that of Pop III stars. They calculated 10 6 binary evolutions for given initial values of the primary mass M 1 , the mass ratio M 2 /M 1 , the orbital separation a and the eccentricity e using the Monte Carlo method under the initial distribution functions. They call the primary star as the larger mass one while the secondary is the smaller mass one in the binary. The typical mass of Pop III stars is from ∼ 10 M ⊙ to ∼ 100 M ⊙ [25,26]. Thus, they took the initial mass function which may be flat from 10 M ⊙ to 140 M ⊙ suggested from the numerical simulations [27,28]. The reason for the upper limit of mass of 140 M ⊙ is that the star with mass larger than 140 M ⊙ becomes the pair instability supernova leaving no remnant. Since there is no observation of Pop III stars and binaries, they simply assume that other initial distribution functions are the same as Pop I binaries. The initial mass ratio function for given M 1 is flat from 10 M ⊙ /M 1 to 1. The separation a 2 distribution function is proportional to 1/a from a min to 10 6 R ⊙ , where a min is the minimum separation when the binary interaction such as the mass transfer and so on is absent. The initial eccentricity distribution function is proportional to e from 0 to 1. The set of these initial distribution functions is the same as their standard model with 140 case of Ref. [22]. In this paper, we choose the binary evolution parameters of their standard model and the optimistic core-merger criterion of Ref. [22]. The details of binary interactions and spin evolution, which is very important in this paper, are discussed in Refs. [21,22].
In Ref. [22], they found that ∼ 13 % of Pop III binaries become BH-BH binaries which merge within the Hubble time and the typical mass of Pop III BHs is ∼ 30 M ⊙ . Figure 1 shows the initial mass ratio distribution of Pop III binaries (red line) and that of Pop III BH-BHs (blue dashed line). Even though the mass ratio of binaries smaller than ∼ 0.5 exists substantially initially, most of BH-BH binaries has mass ratio larger than ∼ 0.5 by the effect of mass transfer. Thus, large mass ratio (= M 2 /M 1 ) BH-BHs are the majority. Figure 2 shows the distribution of spin parameters of the primary and secondary BHs when the primary and secondary become BHs. Figure 3 shows the cross section views of distributribution of spin parameter with (a) the cross section views of distributribution of spin parameter when 0 < q 1 < 0.05 and (b) the cross section views of distributribution of spin parameter when 0.95 < q 1 < 0.998. The spin parameter of each BH is calculated by the angular momentum of the progenitor just before it becomes BH. If the spin parameter of the BH is larger than the Thorne limit [29], we assign q = q Thorne = 0.998 as the spin parameter. From Fig. 2 and Fig. 3, the spin parameters of Pop III BH-BHs are roughly classified into 3 groups. First, the majority of Pop III is in the group where both BHs have high spin parameters. If the mass transfer is dynamically unstable or the secondary plunges into the primary envelope, the orbit shrinks and the primary envelope is stripped by the friction between the secondary and the primary envelope [30]. In this group, the progenitors evolve without the common envelope phase and the primary envelope is not stripped. Thus, BHs of this group get large angular momentum from the envelope of progenitor and the spin parameter which has the largest Thorne limit q Thorne = 0.998. Second, there is a group where both BHs have low spin parameters. In this group, each star evolves via the common envelope phase and they take off their envelope and lose almost all of the angular momentum. Thus, there are many Pop III BH-BHs with q 1 < 0.15, q 2 < 0.15. Third, there is a group where the one of the pair has high spin and the other has low spin. In this group, the primary evolves with the common envelope phase and the secondary evolves without the common envelope, or vice versa.
3. Remnant mass, spin and the detection rate Given BH binary parameters, M 1 , M 2 , q 1 and q 2 , we calculate the remnant mass and spin by using formulae from spin aligned BH binaries [31,32] (see also Ref. [33,34] from a different group). The final (non-dimensional) spin parameter q f is where and (+ · ··) denotes the higher order correction with respect to spins which is given in Eq. (14) of Ref. [32] explicitly. L 0 , L 1 and L 2a are the fitting parameters summarized in Table VI of Ref. [32], and the last two terms in Eq. (4) are added to enforce the particle limit (η → 0) whereJ ISCO is the orbital angular momentum of the innermost stable circular orbit (ISCO). (a) The distribution of q 2 for 0 < q 1 < 0.05. We can see that q 2 distribution has bimodial peaks at 0 < q 2 < 0.15 and 0.95 < q 2 < 0.998 . (b) The distribution of q 2 for 0.95 < q 1 < 0.998. We see that the large value of q 2 is the majority so that there is a group in which both q 1 and q 2 are large.
The final mass M f is given by Again, M 0 , K 1 and K 2a are the fitting parameters summarized in Table VI of Ref. [32]. In practice, we use Ref. [35] for the ISCO angular momentum and energy,J ISCO andẼ ISCO (note that we assign q f to a in Ref. [35]). According to a recent numerical relativity simulation for a highly spinning BH binary merger with M 1 = M 2 = 1/2, q 1 = q 2 = 0.994 [8], the final mass and spin after merger are 5/9 obtained as M f = 0.887 and q f = 0.950, respectively. On the other hand, the remnant formulae in Eqs. (4) and (6) which are not calibrated by the above numerical relativity result, give M f = 0.888 and q f = 0.950. We see that the formulae are sufficiently accurate for our analysis (see also a recent study on remnant BHs for precessing BH binaries [36]). The radiated energy is so large that the total mass of 60 M ⊙ for the above highly spinning binaries becomes the remnant mass of 53.28 M ⊙ . We note that Eq. (4) cannot give any realistic solution for some large mass ratio, e.g., for η 0.1249 (q 1 = q 2 = 0.998), η 0.1169 (q 1 = q 2 = 0.994) and η 0.1066 (q 1 = q 2 = 0.99). In that case, we simply set q f = 0.998.
In Fig. 4, we show the remnant mass and spin calculated by the remnant formulae. Due to the mass decrease by the gravitational wave radiation, we see the peak in a bin between 50 M ⊙ and 60 M ⊙ which reflects the peak of the total mass of Pop III BH-BHs. The remnant spin q f > 0.96 is 1.56%, and only 0.429% of the remnant BHs have the spin larger than 0.98.  To estimate the signal-to-noise ratio (SNR) of the QNM (ringdown) signal in the expected noise curve of KAGRA [13,14] [bKAGRA, VRSE(D) configuration] shown in Ref.
[37], we use the results derived by Flanagan and Hughes in Ref. [38]. In Ref. [39], we have fitted the KAGRA noise curve as S n (f ) 1/2 = 10 −26 6.5 × 10 10 f −8 + 6 × 10 6 f −2.3 + 1.5f where the frequency f is in units of Hz. According to Ref. [38], the angle averaged SNR for the ringdown phase is calculated from Eq. (B14) of Ref. [38] as where F (q f ) and f c are given in Ref. [40], and ǫ r denotes the fraction of the total mass energy radiated in the ringdown phase which is assumed as ǫ r = 0.03. Here, we have ignored effects of the redshifted mass and the cosmological distance, i.e., the redshift and the difference between the distance D and the 6/9 luminosity distance. Since the maximum distance considered here is z ∼ 0.28, the errors are small. Although the calculation is straightforward, we do not show the explicit expression since the expression is complicated due to S n (f c ). For example, we have SNR = 23 in the case of M f = 60 M ⊙ , q f = 0.7, η = 1/4 and D = 200 Mpc. Figure 5 shows the normalized distribution of the SNRs. Here, we have assumed all gravitational wave sources are located at D = 200 Mpc. To calculate the detection rate, we need to know the merger rate density of Pop III BH-BHs. The merger rate density derived in Ref. [22] is approximated by R m = 0.024 + 0.0080 (D/1 Gpc) [Myr −1 Mpc −3 ] for a low redshift (note that this fitting works up to z ∼ 2). In this paper, we focus on the detection rate of the solid angle of a sphere emitting the QNM, 4πC which dips in the ergoregion. In Ref. [15], we obtained a simple relation between C and the spin parameter q shown as Eq. (3) in Introduction. In Table 1, we present the result for the detection rate in the case of SNR = 35 which is needed to confirm the QNM [41].  [42][43][44] showed that the some fraction of Pop II evolves into massive BH-BHs. In addition, the rotating Pop II stars are easier to become massive BH-BHs than non-rotating Pop II stars [45,46]. Since the mass loss is expected for Pop I and Pop II stars due to the absorption of photons at the spectral lines of metals, these Pop I and Pop II cases, however, the progenitors are easier to lose the angular momentum by the stellar wind mass loss and so on. Thus, the Pop I and Pop II cases might have lower spin than the Pop III case. If , however, the mass and the spin distributions of references [44][45][46] are available, we can compute the detection rate of the QNM and the covered solid angle 4πC of the ergoregion. In this sense, the rate shown in this paper is the minimum one. It is noted that from the standard model of Pop III population synthesis, the event rate of the final q f > 0.98 BHs is ∼ 5.17 × 10 −6 events yr −1 (SFR p /(10 −2.5 M ⊙ yr −1 Mpc −3 )) · ([f b /(1 + f b )]/0.33) for SNR = 35. We expect interesting physics in highly/extremely spinning Kerr BHs, which needs further studies. To detect such a BH, third generation gravitational wave detectors, such as the Einstein Telescope (ET) [47] should be required. Also, although there is room for highly spinning remnant BHs with q f > 0.98 from the merger of comparable mass BH binaries, we will need large-mass-ratio binaries for which the cosmic censorship [5] has been discussed extensively (see e.g., Ref. [48] and references therein). These binaries could be one of the targets for space based gravitational wave detectors such as eLISA [49] and DECIGO [50] at the formation time of z ∼ 10.