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Slowly Decaying Ringdown of a Rapidly Spinning Black Hole: Probing the No-Hair Theorem by Small Mass-Ratio Mergers with LISA

The detectability of multiple quasi-normal (QN) modes, including overtones and higher harmonics, with the Laser Interferometer Space Antenna (LISA) is investigated by computing the gravitational wave (GW) signal induced by an intermediate or extreme mass ratio merger involving a supermassive black hole (SMBH). We confirm that the ringdown of rapidly spinning black holes are long-lived, and higher harmonics of the ringdown are significantly excited for mergers of small mass ratios. We demonstrate that the observation of GWs from rapidly rotating SMBHs has a significant advantage for detecting multiple QN modes and testing the no-hair theorem of black holes with high accuracy.


I. INTRODUCTION
We are in a golden age of gravitational-wave (GW) astronomy, where mergers of binary black holes (BHs) are discovered by GW interferometers [1][2][3][4][5][6][7]. The end product of a merger is a distorted single BH, which settles down to a Kerr BH by radiating GWs. This ringdown phase is characterized by a set of damped sinusoids called quasi-normal (QN) modes, and is an important probe to test general relativity in the strong-gravity regime [8][9][10]. QN modes from BH merger remnants have been detected for a large number of events, and were used for various tests of general relativity (e.g., [11][12][13][14]).
QN modes consist of fundamental modes and overtones, where the latter is short-lived but can be important for characterizing the ringdown signal [15,16]. Detection of overtones from ringdowns is important for e.g. tests of the no-hair theorem [17]. Evidence of an overtone was claimed in the ringdown of GW 150914 [18], although its significance is still controversial [19][20][21].
A key parameter that governs the relative strength between fundamental modes and overtones is the spin parameter 1 , j ≡ J/M 2 , of the remnant BH. J is the angular momentum and M is the mass of the BH. Recently one of the authors found [16,22] that remnants with rapid spin (j 0.9) can have a ringdown dominated by higher overtones and higher angular modes. The more QN modes are detected, the more accurate the test of general relativity would be. Therefore GW ringdown of a highly spinning or near-extremal BH may be a preferred signal to test general relativity. However the final spin of observed BH mergers is typically ≈ 0.7 [2,3,5], and such extreme spins may be difficult to probe for mergers of stellar-mass BHs whose natal spins are expected to be rather low [23].
In this work we consider the possibility of exploring overtones and higher angular modes of rapidly spinning BHs with intermediate/extreme mass ratio mergers involving a supermassive BH (SMBH) 2 . These sources, especially with SMBHs in the mass range 10 6 -10 7 M , are targets for space-based GW detectors like the Laser Interferometer Space Antenna (LISA) [24,25]. Notably these SMBHs are predicted to have large spins due to gas accretion upon their growth ( [26][27][28], but see [29]). Although systematic uncertainties may exist in the fitting, X-ray spectra of local SMBHs in this mass range indicate high spins of > 0.9, consistent with this scenario [30][31][32].
Using the waveform modelling of a particle plunging into a rapidly spinning BH and extracting the excited QN modes with a fitting analysis, we estimate the detectability of multiple QN modes, including higher overtones and higher angular modes, by LISA 3 . We find that these modes can be detectable out to cosmological distances, realizing a novel probe of gravity in the nearextreme Kerr spacetime.

II. RINGDOWN & BH SPECTROSCOPY
We numerically compute the GW signal induced by a particle plunging into a rotating hole using the Sasaki-Nakamura (SN) equation [34]: where X lm is a perturbation variable of the gravitational field, r * is the tortoise coordinate, F lm and U lm are func- tions with explicit form given in Ref. [34], andT lm is the source term associated with the plunging particle. The form ofT lm is given in Ref. [35] and can be obtained from the geodesic motion of the object. The orbital angular momentum of the plunging orbit is µ × L z , and the infalling condition is −2M . We here assume that the value of L z for infalling objects is typically O(M ) and take L z = 2M throughout the manuscript 4 . Integers l and m are respectively the angular and azimuthal numbers of the spheroidal harmonics. We here consider a situation where the trajectory of a compact object of mass µ is restricted to the equatorial plane 5 (θ = π/2) and the total energy of the object (including rest energy) is µ. The self force of the object can be neglected, which is valid for a small mass ratio q ≡ µ/M 1. Using the Green's function technique, one can solve the SN equation as where G(r , r * , ω) is the Green's function that is obtained from the homogeneous solution of the SN equation. We then obtain the GW spectrum where a ≡ J/M and −2 S lm is the spin-weighted spheroidal harmonics, assuming an edge-on observer with argument π/2. The time-domain data h = h + + ih × = (l,m) h lm is obtained by the Fourier transformation of h. The function R lm (ω) is obtained by properly normalizing X lm , and its explicit form is provided in Ref. [34].
The GW spectra computed with this scheme is shown in Figure 1. One can see that (l, m) = (2, 2) dominates the GW signal for an intermediate spin (j = 0.8 in Figure  1), but higher angular modes are significantly excited for a near-extremal Kerr BH of j = 0.99. We are interested in the signal induced by a compact object plunging into a rapidly spinning BH, as more QN modes are long-lived for higher spins ( Figure 2).
In the next Section, we show that a number of highly damped modes dominate the early ringdown of a nearextremal BH by fitting multiple overtones and fundamental modes to the GW signal. Then we show that rapidly spinning BHs are better targets to perform a high-precision detection of multiple QN modes, including higher angular modes and higher overtones.

III. EXCITATION OF OVERTONES
The detectability of the QN modes is highly sensitive to the start time of ringdown t * because (i) it is a superposition of QN modes, each of which is exponentially damped in time, and (ii) overtones may dominate the signal at early times. The exact start time of the ringdown is unknown, but one can obtain a best fit value by fitting multiple QN modes to the GW signal. We then show that the ringdown starts earlier than the strain peak, around the time when the object plunges into the photon sphere.
We perform fitting analysis of QN modes 6 including overtones of n ≤ n max and angular modes of 2 ≤ l = m ≤ 5. The fitting is done in the frequency domain, to avoid the instability at early ringdown [39] and to treat t * as one of the fitting parameters [40]. Figure 3-(a) shows that the best fit value of t * (in units of 2M ) is in the range 40 t * 43, where the mismatch M is less than 0.01. The range of t * with M < 0.1 corresponds to the moments after the particle plunges into the photon sphere (see Figure 3-(b)). This is consistent with the fact that GW ringdown is a signal associated with the photon sphere [41]. We also find that including up to higher overtones (n max 15) in the fit is necessary to guarantee the convergence of the mismatch ( Figure  4). That is, once we admit the start time of ringdown is soon after the compact object plunges into the photon sphere (t * ∼ 42 in our setup), not only the fundamental modes with quadrupole moment but also a number of long-lived modes and higher harmonics are significantly excited for a near-extremal BH. It has an advantage for measuring multiple QN modes, including overtones and higher harmonics, and for accurately testing general relativity. Indeed, it is reported that a fraction of SMBHs can have near-extremal spin parameters of j 0.99 [30][31][32]. In the following, we show such SMBHs are suitable observation targets to detect multiple QN modes.

IV. NO-HAIR TEST OF SMBHS WITH LISA
Let us evaluate the detectability of multiple QN modes of rotating BHs, and its feasibility for tests of the nohair theorem. We here consider detections of them with LISA, which is sensitive to signals of ∼ 0.01 Hz and is suitable for detecting ringdown of SMBHs with mass ∼ 10 6 -10 7 M . The sensitivity curve can be modeled by an analytic functionS n (f ) [42]: where 2L is the round-trip light travel distance, f * ≡ c/(2πL) is the transfer frequency, P OMS and P acc are the single-link optical metrology noise and the single test mass acceleration noise, respectively. The function F is the sky/polarization average of the antenna pattern functions 7 . We include the galactic confusion noise from compact binaries, where A = 9×10 −45 and the parameter set {α, β, γ, κ, f k } is fixed with the values for a 4-year mission (we use Table 1 in Ref. [42]). Then we obtain the full sensitivity curve as S n (f ) =S n (f ) + S c (f ). Using S n , we evaluate the SNR of the GW signal and the Bayes factor, to see the support for the model of the no-hair QN modes H 0 over a modified model H δ consisting of a set of complex frequencies deviated from the no-hair values.
For the modification of the no-hair model, we consider two types of modifications: (i) a model H where H δ is H +/× (f ) is the Fourier transform of h +/× (t), and p(d|H) is the likelihood with a given data, d, and a model function, H( ϑ), for a set of the fitting parameters ϑ, i.e., the amplitude and phase of each QN mode and t * . Figure 5 shows the precision of constraining deviations of multiple QN modes for different thresholds of B. We find that no-hair tests of rapidly rotating BHs are more powerful than those with BHs of intermediate spin. For the H (F+O) δ model in the left panel, the measurement error of the real frequency is 80% for j = 0.8, but only 10% for j = 0.99. We here take (D L ,M , q) = (3 Gpc, 10 7 M , 10 −3 ), where D L is the luminosity distance andM = (1 + z)M is the redshifted mass of a SMBH that includes the effect of redshift z.
Overtones more quickly damp with time, and hence measuring deviations of overtones are likely more challenging. Nevertheless, for near-extremal BHs, the damping is very weak and even overtones may be measured with good precision. The right panel in Figure  5 shows the high feasibility of measuring overtones for a rapidly spinning BH. We here assume (D L ,M , q) = (1 Gpc, 10 7 M , 10 −3 ).
In both models that we considered, the uncertainty in the damping rate δf I of QN modes is larger. A similar conclusion was reported in [18], where the no-hair test was performed for GW150914 [44]. Figure 6 summarizes the expected distance out to which we can measure multiple QN modes with high precision. In this section we discuss the prospects for LISA, for moderate and extreme mass-ratios.

V. DISCUSSION
For moderate mass ratios of q ∼ 10 −3 , it corresponds to a merger between a SMBH and an intermediate-mass BH (IMBH). A scenario usually considered for such mergers is clusters hosting IMBHs falling into the galactic nuclei [45][46][47][48][49]. The event rate is uncertain, but recent N-body simulations find a range 0.003-0.03 Gpc −3 yr −1 [48,49], or 2-20 yr −1 within z < 1 (D L 7 Gpc) [25]. For  with δf R > 0 is more significant than that with δf R < 0 (see Figure  6). Being sensitive to the modification of δf R > 0 is reasonable since the higher-frequency modes of ω ω lmn FIG. 6: Distances out to which tests with multiple QN modes are possible. The contours show the SNR of the ringdown, as a function of the redshifted SMBH massM and the luminosity distance DL with mass ratio q scaled as 10 −3 . The solid and dashed lines indicate δfR > 0 and δfR < 0, respectively. In the shaded regions, below both the solid and dashed lines, one can measure the real part of QN frequencies with (green) 1%, (red) 5%, and (blue) 10%, respectively. We here set δfI = 0, as observations are rather insensitive to δfI (see Figure 5).
in the GW signal are exponentially suppressed (see Ref. [22] for more details).
For extreme mass ratios of q < 10 −5 , it corresponds to a stellar-mass BH plunging into a SMBH. Such plunges are expected not to be strong GW emitters, as we also deduce from Figure 6. The EMRI rate for a Milky-Way like Galaxy is estimated to be 10 −6 -10 −5 yr −1 , i.e. an event rate of 10 −8 -10 −7 Mpc −3 yr −1 ([50] and references therein). Plunge orbits can be up to 100 times more likely than EMRIs [51,52], so we expect plunges of ∼ 10 M BHs within the detectable distance ( 10 Mpc) at a rate of < 0.1 yr −1 .
Recently a new formation channel of IMBHs in galactic nuclei was proposed, where stellar-mass BHs grow in-situ up to ∼ 10 4 M by collisions with surrounding stars [53]. If such growth is efficient, this would likely enhance the above rates.
In summary, we demonstrated that intermediate (and possibly extreme) mass ratio mergers are ideal targets for LISA to probe multiple QN modes of rapidly spinning BHs. This may become a unique target for tests of gravity in a near-extreme Kerr spacetime.