RIKEN-iTHEMS-Report-22 RESCEU-1922 YITP-23-155 Slowly Decaying Ringdown of a Rapidly Spinning Black Hole Probing the No-Hair Theorem by Small Mass-Ratio Mergers with LISA

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RIKEN-iTHEMS-Report-22, RESCEU-19/22, YITP-23-155

Slowly Decaying Ringdown of a Rapidly Spinning Black Hole:

Probing the No-Hair Theorem by Small Mass-Ratio Mergers with LISA

Naritaka Oshita1,2,3∗and Daichi Tsuna4,5†

1RIKEN iTHEMS, Wako, Saitama, 351-0198, Japan

2Center for Gravitational Physics and Quantum Information,

Yukawa Institute for Theoretical Physics, Kyoto University,

Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan

3The Hakubi Center for Advanced Research, Kyoto University,

Yoshida Ushinomiyacho, Sakyo-ku, Kyoto 606-8501, Japan

4Research Center for the Early Universe (RESCEU),

Graduate School of Science, The University of Tokyo,

7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan and

5TAPIR, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA

The measurability of multiple quasinormal (QN) modes, including overtones and higher harmon-

ics, with the Laser Interferometer Space Antenna is investigated by computing the gravitational

wave (GW) signal induced by an intermediate or extreme mass ratio merger involving a supermas-

sive 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

investigate the measurability and separability of the QN modes for such mergers and demonstrate

that the observation of GWs from rapidly rotating SMBHs has an advantage for detecting super-

posed QN modes and testing the no-hair theorem of black holes.

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 [1–7]. 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

quasinormal (QN) modes, and is an important probe to

test general relativity in the strong-gravity regime [8–10].

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., [11–14]).

QN modes consist of fundamental modes and over-

tones, where the latter is short-lived but can be impor-

tant for characterizing the ringdown signal [15,16]. De-

tection of overtones from ringdowns is important for e.g.

tests of the no-hair theorem [17]. Evidence of an over-

tone was claimed in the ringdown of GW 150914 [18],

although its signiﬁcance is still controversial [19–21].

A key parameter that governs the relative strength be-

tween fundamental modes and overtones is the spin pa-

rameter,1j≡J/M2, of the remnant BH. Jis the angular

momentum and Mis 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 over-

tones and higher angular modes. The more QN modes

are detected, the more accurate the test of general rel-

ativity would be. Therefore GW ringdown of a highly

∗Electronic address: naritaka.oshita@yukawa.kyoto-u.ac.jp

†Electronic address: tsuna@caltech.edu

1In this work we use the natural units c=ℏ= 1 and G= 1.

spinning or near-extremal BH may be a preferred signal

to test general relativity. However the ﬁnal spin of ob-

served 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 in-

volving a supermassive BH (SMBH).2These sources, es-

pecially with SMBHs in the mass range 106–107M⊙,

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 ([26–28], but see [29]). Al-

though 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 [30–32].

Using the waveform modeling of a particle plunging

into a rapidly spinning BH and extracting the excited QN

modes with a ﬁtting analysis, we estimate the measura-

bility of multiple QN modes, including higher overtones

and higher angular modes, by LISA.3We ﬁnd that these

modes can be detectable out to cosmological distances,

realizing a novel probe of gravity in the near-extreme

Kerr spacetime. We also evaluate the error of the mea-

2The self-force of the plunging object is ignored in our computa-

tion, as we consider the orbit of a light object. In other words,

dephasing of GWs and the backreaction of the object to the tra-

jectory are assumed to be subdominant.

3For the LISA detectability of the fundamental mode and the

ﬁrst overtone whose amplitude is assumed to be 1/10 that of the

fundamental one, see Ref. [33].

arXiv:2210.14049v2 [gr-qc] 30 Nov 2023

FIG. 1: Spectra of the GW signal induced by a compact object plunging into a BH with Lz= 1 and two BH spins, j= 0.8

(left) and 0.99 (right). We adopt the normalization 2M= 1.

surability and separability of the QN modes [34,35] to

assess the feasibility of measuring individual modes.

II. RINGDOWN FOR A SMALL MASS RATIO

MERGER

In this work we focus on simulating a merger of small

mass ratio, such that its dynamics can be well approxi-

mated by a test particle plunging into a SMBH. We nu-

merically compute the GW signal induced by a particle

plunging into a rotating hole using the Sasaki-Nakamura

(SN) equation [36]:

d2

dr∗2−Flm

dr∗−UlmXlm =˜

Tlm,(1)

where Xlm is a perturbation variable of the gravitational

ﬁeld, r∗is the tortoise coordinate, Flm and Ulm are func-

tions with explicit form given in Ref. [36], and ˜

Tlm

is the source term associated with the plunging parti-

cle.4The form of ˜

Tlm is given in Ref. [37] and can

be obtained from the geodesic motion of the object.

The orbital angular momentum of the plunging orbit is

µ×Lz, and the infalling condition is −2M(1+√1 + j)<

Lz<2M(1 + √1−j). We here assume that the value

of Lzfor infalling objects is typically O(M) and take

Lz= 2Mthroughout the manuscript.5Integers land 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 µis re-

4To simulate a particle plunging from a ﬁnite distance from the

SMBH (not from inﬁnity as was assumed in Ref. [37]), we mod-

iﬁed the source term in Ref. [37]. We suppress the contribution

of the source term at ω≪1/M including at ω= 0 (originating

from the particle motion at inﬁnity), by multiplying f0and f1

in the source term in Appendix B in Ref. [37] by 2Mω.

5We plan to investigate the dependence of GW signals on Lzin

a forthcoming paper. Note that when |Lmax/min −Lz|/M ≪1,

where Lmax and Lmin are, respectively, the upper and lower

limits of Lz, the object follows a circulating orbit and the self-

force would not be negligible.

stricted to the equatorial plane6(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

lim

r∗→∞ Xlm(ω, r∗) = X(out)

lm (ω)eiωr∗

=Zdr′˜

Tlm(r′, ω)G(r′, r∗, ω),

(2)

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

h=X

l,m

hlm(ω) = X

l,m −2

ω2−2Slm(aω, π/2)Rlm(ω),(3)

where a≡J/M and −2Slm is the spin-weighted

spheroidal harmonics, assuming an edge-on observer with

argument π/2. The time-domain data h=h++ih×=

FIG. 2: Kerr QN frequencies for (l, m) = (2,2), with the nor-

malization 2M= 1. Diﬀerent markers correspond to diﬀerent

BH spins, and nincreases from top to bottom of this plot. For

a higher spin, an increasing number of overtones have small

Im(ωlmn) and are long-lived.

6The Carter constant, one of the parameters characterizing tra-

jectories around BHs, is set to zero in our computation.

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RIKEN-iTHEMS-Report-22,RESCEU-19/22,YITP-23-155SlowlyDecayingRingdownofaRapidlySpinningBlackHole:ProbingtheNo-HairTheorembySmallMass-RatioMergerswithLISANaritakaOshita1,2,3∗andDaichiTsuna4,5†1RIKENiTHEMS,Wako,Saitama,351-0198,Japan2CenterforGravitationalPhysicsandQuantumInformation,YukawaInstitutefo...

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RIKEN-iTHEMS-Report-22 RESCEU-1922 YITP-23-155 Slowly Decaying Ringdown of a Rapidly Spinning Black Hole Probing the No-Hair Theorem by Small Mass-Ratio Mergers with LISA

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