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Multipolar gravitational waveforms and ringdownsgenerated during the plunge from the innermost stable

circular orbit into a Schwarzschild black holeAntoine Folacci, Mohamed Ould El Hadj

To cite this version:Antoine Folacci, Mohamed Ould El Hadj. Multipolar gravitational waveforms and ringdowns gen-erated during the plunge from the innermost stable circular orbit into a Schwarzschild black hole.Phys.Rev.D, 2018, 98 (8), pp.084008. �10.1103/PhysRevD.98.084008�. �hal-01817920�

https://hal.archives-ouvertes.fr/hal-01817920https://hal.archives-ouvertes.fr

Multipolar gravitational waveforms and ringdowns generatedduring the plunge from the innermost stable circular

orbit into a Schwarzschild black hole

Antoine Folacci* and Mohamed Ould El Hadj†

Equipe Physique Théorique, SPE, UMR 6134 du CNRS et de l’Université de Corse,Université de Corse, Faculté des Sciences, BP 52, F-20250 Corte, France

(Received 5 June 2018; published 5 October 2018)

We study the gravitational radiation emitted by a massive point particle plunging from slightly below theinnermost stable circular orbit into a Schwarzschild black hole. We consider both even- and odd-parityperturbations and describe them using the two gauge-invariant master functions of Cunningham, Price, andMoncrief. We obtain, for arbitrary directions of observation and, in particular, outside the orbital plane ofthe plunging particle, the regularized multipolar waveforms, i.e., the waveforms constructed by summingover of a large number of modes, and their unregularized counterparts constructed from the quasinormal-mode spectrum. They are in excellent agreement and our results permit us to especially emphasize theimpact on the distortion of the waveforms of (i) the harmonics beyond the dominant (l ¼ 2, m ¼ �2)modes and (ii) the direction of observation, and therefore the necessity to take them into account in theanalysis of the last phase of binary black hole coalescence.

DOI: 10.1103/PhysRevD.98.084008

I. INTRODUCTION

In this article, we shall obtain and analyze in terms ofquasinormal modes (QNMs) the multipolar gravitationalwaveforms generated by a massive “point particle” plung-ing from slightly below the innermost stable circular orbit(ISCO) into a Schwarzschild black hole (BH). Here, it isimportant to note that, by multipolar waveforms, we intendwaveforms constructed by superposition of a large numberof modes. We shall assume an extreme mass ratio for thephysical system considered, i.e., that the BH is muchheavier than the particle, such a hypothesis permitting us todescribe the emitted radiation in the framework of BHperturbations [1–7]. In the context of gravitational wavephysics and with the first direct gravitational-detection of abinary black hole coalescence by LIGO [8], the problem westudy is of fundamental importance and there exists a largeliterature concerning it more or less directly (see, e.g.,Refs. [9–27]). Indeed, the “plunge regime” from the ISCOis the last phase of the evolution of a stellar mass objectorbiting near a supermassive BH or it can be also used todescribe the late-time evolution of a binary BH. Here, it isimportant to recall that, as a result of the radiation ofgravitational waves, the eccentricity and the semimajor axisof a two-point mass system decay [28] and, as a conse-quence, for a wide class of initial conditions at largedistances, the orbits are “circularized” and the system

reaches an ISCO or an effective ISCO. Moreover, thewaveform generated during the plunge regime encodesthe final BH fingerprint. It should be recalled that, in thiscontext, a multipolar description of the gravitational signalwill be of fundamental interest with the enhancementof the sensitivity of laser-interferometric gravitational wavedetectors (see, e.g., Refs. [29–31] and references therein).In this article, which generalize our recent work concerningthe electromagnetic radiation emitted by a charged particleplunging from the ISCO into a Schwarzschild black hole[32], we shall show, by taking into account a large numberof higher harmonics, that the waveform is strongly distortedand that the distortion highly depends on the direction ofobservation.It should be pointed out that our work extends the study

of Hadar and Kol [19] as well as the analysis of Hadar, Kol,Berti, and Cardoso in Ref. [20]. In these two articles, themultipolar and quasinormal BH responses have beentheoretically constructed for any angular position of theobserver but the corresponding numerical aspects are ratherlimited. For instance, in Ref. [19], the authors have onlyplotted the quasinormal response observed in the orbitalplane of the plunging particle. We go out the orbital plane inour numerical results and plots. We also go further byplotting and comparing, in the same figures and for manyangular positions of the observer, the multipolar gravita-tional waveforms and the quasinormal ringdowns. InRefs. [19,20], the multipolar waveforms are never plottedand the comparison of the partial waveforms obtained inRef. [20] with the quasinormal ringdowns constructed in

*[email protected]†[email protected]

PHYSICAL REVIEW D 98, 084008 (2018)

2470-0010=2018=98(8)=084008(22) 084008-1 © 2018 American Physical Society

https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevD.98.084008&domain=pdf&date_stamp=2018-10-05https://doi.org/10.1103/PhysRevD.98.084008https://doi.org/10.1103/PhysRevD.98.084008https://doi.org/10.1103/PhysRevD.98.084008https://doi.org/10.1103/PhysRevD.98.084008

Ref. [19] is achieved in an alternative way based on anumerical fitting method. Moreover, it is important to notethat, due to a sign difference, our results concerning themultipolar quasinormal waveforms observed in the orbitalplane of the plunging particle do not agree with thenumerical results plotted in Ref. [19].Our paper is organized as follows. In Sec. II, after a brief

overview of gravitational perturbation theory in theSchwarzschild spacetime, we establish theoretically theexpression of the waveforms emitted by a massive pointparticle plunging from the ISCO into the BH. Moreprecisely, we consider both the even- and odd-paritygravitational perturbations for arbitrary (l, m) modesand describe them using the two gauge-invariant masterfunctions of Cunningham, Price, and Moncrief [3–7]. Wethen solve, in the frequency domain and by using standardGreen’s function techniques, the Regge-Wheeler equation[1] governing the odd-perturbations as well as the Zerilli-Moncrief equation [2,3] governing the even-perturbations.This is achieved after having constructed the sources forthese two equations from the closed-form expression of theplunge trajectory. In Sec. III, we extract from the results ofSec. II the QNM counterpart of the waveforms correspond-ing to the gravitational ringing (or ringdown) of the BH. Wegather all our numerical results and their analysis in Sec. IVwhere we display the regularized multipolar waveformsemitted [i.e., the waveforms constructed by summing overof a large number of (l, m) modes] and compare them withtheir unregularized counterparts constructed solely from theQNM spectrum. Both are obtained for arbitrary directionsof observation and, in particular, outside the orbital plane ofthe plunging particle. It should be noted that, in the latephase of the signals, they are in excellent agreement.Moreover, our results especially emphasize the impacton the distortion of the waveforms of the harmonics beyondthe dominant (l ¼ 2, m ¼ �2) modes and of the directionof observation. At the end of this section, we also compareour results with those displayed in Ref. [19] and propose anexplanation for the disagreement found. In the Conclusion,we briefly summarize the main results obtained in thisarticle and, in an Appendix, we carefully examine theregularization of the partial amplitudes from both thetheoretical and numerical point of view. Indeed, the exactwaveforms theoretically obtained in Sec. II are integralsover the radial Schwarzschild coordinate which arestrongly divergent near the ISCO. For even as well asfor odd perturbations, they can be numerically regularizedby using the Levin’s algorithm [33] but only after havingreduced the degree of divergence of these integrals by asuccession of integrations by parts, i.e., by extending themethod we developed in our work concerning the chargedparticle plunging from the ISCO into a Schwarzschild blackhole where we encountered a similar problem [32].Throughout this article, we adopt units such that G ¼

c ¼ 1 and we use the geometrical conventions of Ref. [34].

II. GRAVITATIONAL WAVES GENERATED BYTHE PLUNGING MASSIVE PARTICLE

In this section, we shall obtain theoretically the expres-sion of the even- and odd-parity waveforms emitted by amassive point particle plunging from slightly below theISCO into the BH. This will be achieved by working in thefrequency domain and using the standard Green’s functiontechniques. Moreover, we shall fix the notations andconventions used throughout the whole article.

A. The Schwarzschild BH and theplunging massive particle

We recall that the exterior of the Schwarzschild BH ofmass M is defined by the metric

ds2 ¼ −fðrÞdt2 þ fðrÞ−1dr2 þ r2dσ22 ð1Þ

where fðrÞ ¼ ð1 − 2M=rÞ and dσ22 ¼ dθ2 þ sin2θdφ2denotes the metric on the unit 2-sphere S2 and with theSchwarzschild coordinates (t, r, θ, φ) which satisfyt ∈� −∞;þ∞½, r ∈�2M;þ∞½, θ ∈ ½0; π� and φ ∈ ½0; 2π�.In the following, we shall also use the so-called tortoisecoordinate r� ∈� −∞;þ∞½ defined in terms of the radialSchwarzschild coordinate r by dr=dr� ¼ fðrÞ and given byr�ðrÞ ¼ rþ 2M ln½r=ð2MÞ − 1�. We recall that the functionr� ¼ r�ðrÞ provides a bijection from �2M;þ∞½ to� −∞;þ∞½.We denote by tpðτÞ, rpðτÞ, θpðτÞ and φpðτÞ the coor-

dinates of the timelike geodesic γ followed by the plungingparticle (here τ is the proper time of the particle) and by m0its mass. Without loss of generality, we can consider that itstrajectory lies in the BH equatorial plane, i.e., we assumethat θpðτÞ ¼ π=2. The geodesic equations defining γ aregiven by [35]

fðrpÞdtpdτ

¼ Ẽ; ð2aÞ

r2pdφpdτ

¼ L̃ ð2bÞ

and

�drpdτ

�2

þ L̃2

r2pfðrpÞ −

2Mrp

¼ Ẽ2 − 1: ð2cÞ

Here Ẽ and L̃ are, respectively, the energy and angularmomentum per unit mass of the particle which are twoconserved quantities given on the ISCO, i.e., at r ¼ rISCOwith

rISCO ¼ 6M; ð3Þ

by

ANTOINE FOLACCI and MOHAMED OULD EL HADJ PHYS. REV. D 98, 084008 (2018)

084008-2

Ẽ ¼ 2ffiffiffi2

p

3and L̃ ¼ 2

ffiffiffi3

pM: ð4Þ

By substituting (4) into the geodesic equations (2a)–(2c),we obtain after integration

tpðrÞ2M

¼ 2ffiffiffi2

p ðr − 24MÞ2Mð6M=r − 1Þ1=2 − 22

ffiffiffi2

ptan−1½ð6M=r − 1Þ1=2�

þ 2tanh−1�1ffiffiffi2

p ð6M=r − 1Þ1=2�þ t02M

ð5Þ

and

φpðrÞ ¼ −2

ffiffiffi3

p

ð6M=r − 1Þ1=2 þ φ0 ð6Þ

where t0 and φ0 are two arbitrary integration constants.From (6), we can write the spatial trajectory of the plungingparticle in the form

rpðφÞ ¼6M

½1þ 12=ðφ − φ0Þ2�: ð7Þ

We have displayed this trajectory in Fig. 1.

B. Gravitational perturbations inducedby the plunging particle

Gravitational waves emitted from the Schwarzschild BHexcited by the plunging particle can be characterized by thefield hμν which satisfies the wave equation

□hμν−hρμ;νρ−hρν;μρþh;μνþgμνðhρσ ;ρσ −□hÞ¼−16πTμνð8Þ

where Tμν, which is the stress-energy tensor associated withthe massive particle, is given by

TμνðxÞ ¼ m0Zγdτ

dxμpðτÞdτ

dxνpðτÞdτ

δ4ðx − xpðτÞÞffiffiffiffiffiffiffiffiffiffiffiffi−gðxÞp ð9aÞ

¼ m0dxμpdτ

ðrÞ dxνp

dτðrÞ

�drpdτ

ðrÞ�−1

×δ½t − tpðrÞ�δ½θ − π=2�δ½φ − φpðrÞ�

r2 sin θ: ð9bÞ

In this last equation, tpðrÞ and φpðrÞ are respectivelygiven by (5) and (6).The resolution of the problem defined by (8) and (9b)

and, more generally, the topic of gravitational perturbationsof BHs, have been the subject of lots of works sincethe pioneering articles by Regge and Wheeler [1] andZerilli [2]. So, because gravitational perturbations of theSchwarzschild BH are very well described in the article byMartel and Poisson [6] as well as in the review by Nagarand Rezzolla [7], we just briefly recall some of the resultswe need for our particular work. The gravitational signalemitted can be described in terms of the two gauge-invariant master functions of Cunningham, Price, andMoncrief [3–5] denoted by ψ ðeÞlmðt; rÞ and ψ ðoÞlmðt; rÞ [here,and in the following, the symbols (e) and (o) are respectivelyassociated with even (polar) and odd (axial) objects accord-ing they are of even or odd parity in the antipodal trans-formation on the unit 2-sphereS2] which satisfy respectivelythe Zerilli-Moncrief and Regge-Wheeler equations

�−

∂2∂t2 þ

∂2∂r2� − V

ðe=oÞl ðrÞ

�ψ ðe=oÞlm ðt; rÞ ¼ Sðe=oÞlm ðt; rÞ: ð10Þ

Here the Zerilli-Moncrief potential is given by

VðeÞl ðrÞ¼ fðrÞ

×

�Λ2ðΛþ2Þr3þ6Λ2Mr2þ36ΛM2rþ72M3

ðΛrþ6MÞ2r3�ð11Þ

and we have for the Regge-Wheeler potential

VðoÞl ðrÞ ¼ fðrÞ�Λþ 2r2

−6Mr3

�: ð12Þ

In Eqs. (11) and (12), we have introduced

Λ ¼ ðl − 1Þðlþ 2Þ ¼ lðlþ 1Þ − 2: ð13Þ

0

15 °

30 °

45 °

60 °

75 °90 °105 °

120 °

135 °

150 °

165 °

180 °

195 °

210 °

225 °

240 °

255 °270 ° 285 °

300 °

315 °

330 °

345 °

2M

3M

6M

FIG. 1. The plunge trajectory obtained from Eq. (7). Here, weassume that the particle starts at r ¼ rISCOð1 − ϵÞ with ϵ ¼ 10−3and we take φ0 ¼ 0. The red dashed line at r ¼ 6M and the reddot-dashed line at r ¼ 2M represent the ISCO and the horizon,respectively, while the black dashed line corresponds to thephoton sphere at r ¼ 3M.

MULTIPOLAR GRAVITATIONAL WAVEFORMS AND … PHYS. REV. D 98, 084008 (2018)

084008-3

We note that only the functions ψ ðe=oÞlm ðt; rÞ with l ¼2; 3; 4;… and m ¼ −l;−lþ 1;…;þl are physically rel-evant for our study.We recall that the functions SðeÞlmðt; rÞ and SðoÞlmðt; rÞ

are source terms which depend on the components, inthe basis of tensor spherical harmonics, of the stress-tensor inducing the perturbations of the Schwarzschildspacetime. Their expressions can be found in the

review by Nagar and Rezzolla: SðeÞlmðt; rÞ is given byEq. (4) of the Erratum of Ref. [7] while SðoÞlmðt; rÞ isgiven by Eq. (24) of Ref. [7]). After having checkedthese two results, we have used them to construct thesources corresponding to the stress-energy tensor (9b).By using the orthonormalization properties of the(scalar, vector and tensor) spherical harmonics [6,7],we have obtained

SðeÞlmðt; rÞ ¼8πm0½Ylmðπ=2; 0Þ��ffiffiffiffiffiffi2π

p ðΛþ 2ÞðΛrþ 6MÞ fðrÞ��

3Λ − 2 − 64MΛrþ6M þ 72M2ðΛþ2−m2Þ

r2 þ216M3ðΛþ2−2m2Þ

Λr3

ð6M=r − 1Þ3=2

−8

ð6M=r − 1Þ5=2 − im4

ffiffiffi3

pM

r

�1þ 8ð6M=r − 1Þ3

��δ½t − tpðrÞ�

−12

ffiffiffi2

p ðr2 þ 12M2Þrð6M=r − 1Þ3 δ

0½t − tpðrÞ��exp½−imφpðrÞ� ð14Þ

and

SðoÞlmðt; rÞ ¼16πm0½Xlmφ ðπ=2; 0Þ��ffiffiffiffiffiffi

2πp

ΛðΛþ 2Þ fðrÞ��

−8

ffiffiffi6

pM

r2ð6M=r − 1Þ3=2 þ36

ffiffiffi6

pM2

r3ð6M=r − 1Þ5=2 þ im72

ffiffiffi2

pM2

r3ð6M=r − 1Þ3�δ½t − tpðrÞ�

þ 18ffiffiffi3

pMðr2 þ 12M2Þ

r3ð6M=r − 1Þ3 δ0½t − tpðrÞ�

�exp½−imφpðrÞ�: ð15Þ

In the last equation, we have introduced the vectorspherical harmonic

Xlmφ ¼ − sin θ∂∂θ Y

lm: ð16Þ

Furthermore, we note that the coefficients Ylmðπ=2; 0Þ andXlmφ ðπ=2; 0Þ appearing respectively in Eqs. (14) and (15)are given by

Ylmðπ=2;0Þ¼ 2mffiffiffiπ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2lþ14π

ðl−mÞ!ðlþmÞ!

s

×Γ½l=2þm=2þ1=2�Γ½l=2−m=2þ1� cos ½ðlþmÞπ=2�;

ð17Þand

Xlmφ ðπ=2;0Þ¼2mþ1ffiffiffi

πp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2lþ14π

ðl−mÞ!ðlþmÞ!

s

×Γ½l=2þm=2þ1�Γ½l=2−m=2þ1=2�sin ½ðlþmÞπ=2�: ð18Þ

Here, it is important to remark that Ylmðπ=2; 0Þ and hencethe source (14) vanish for lþm odd while Xlmφ ðπ=2; 0Þand hence the source (15) vanish for lþm even.

It is also important to recall that the partial amplitudes

ψ ðe=oÞlm ðt; rÞ of Cunningham, Price and Moncrief permit usto obtain the gravitational wave amplitude observed atspatial infinity (i.e., for r → þ∞). In the transverse trace-less gauge [34], the two circularly polarized components(hþ, h×) of the emitted gravitational wave are given by [7]

hþ ¼ hðeÞþ þ hðoÞþ and h× ¼ hðoÞ× þ hðoÞ× ð19Þ

with

hðeÞþ ¼1

r

Xþ∞l¼2

Xþlm¼−l

ψ ðeÞlm

�2∂2∂θ2 þ lðlþ 1Þ

�Ylm; ð20aÞ

hðeÞ× ¼ 1rXþ∞l¼2

Xþlm¼−l

ψ ðeÞlm

�2

sin θ

� ∂2∂θ∂φ −

cos θsin θ

∂∂φ

��Ylm;

ð20bÞ

hðoÞþ ¼1

r

Xþ∞l¼2

Xþlm¼−l

ψ ðoÞlm

�2

sin θ

� ∂2∂θ∂φ −

cos θsin θ

∂∂φ

��Ylm;

ð20cÞ


084008-4

hðoÞ× ¼ 1rXþ∞l¼2

Xþlm¼−l

ψ ðoÞlm

�−�2∂2∂θ2 þ lðlþ 1Þ

��Ylm:

ð20dÞ

It should be noted that, due to Eqs. (17) and (18) [seealso the remark following these equations], we have to onlyconsider the couples (l, m) with lþm even in thesuperpositions (20a) and (20b) and the couples (l, m)with lþm odd in the superpositions (20c) and (20d).

C. Construction of the partial amplitudes ψðe=oÞlm ðt;rÞIn order to solve the Zerilli-Moncrief and Regge-

Wheeler equations (10), we shall work in the frequencydomain by writing

ψ ðe=oÞlm ðt; rÞ ¼1ffiffiffiffiffiffi2π

pZ þ∞−∞

dωψ ðe=oÞωlm ðrÞe−iωt ð21Þ

and

Sðe=oÞlm ðt; rÞ ¼1ffiffiffiffiffiffi2π

pZ þ∞−∞

dωSðe=oÞωlm ðrÞe−iωt: ð22Þ

Then, these two wave equations reduce to

�d2

dr2�þ ω2 − VlðrÞ

�ψ ðe=oÞωlm ðrÞ ¼ Sðe=oÞωlm ðrÞ ð23Þ

where the new source terms, which are obtained from (14)and (15), are given by

SðeÞωlmðrÞ ¼8πm0½Ylmðπ=2; 0Þ��ffiffiffiffiffiffi2π

p ðΛþ 2ÞðΛrþ 6MÞ fðrÞ�iω

12ffiffiffi2

p ðr2 þ 12M2Þrð6M=r − 1Þ3 − im

4ffiffiffi3

pM

r

�1þ 8ð6M=r − 1Þ3

�

−8

ð6M=r − 1Þ5=2 þ3Λ − 2 − 64MΛrþ6M þ 72M

2ðΛþ2−m2Þr2 þ

216M3ðΛþ2−2m2ÞΛr3

ð6M=r − 1Þ3=2�exp½iðωtpðrÞ −mφpðrÞÞ� ð24Þ

and

SðoÞωlmðrÞ ¼16πm0½Xlmφ ðπ=2; 0Þ��ffiffiffiffiffiffi

2πp

ΛðΛþ 2Þ fðrÞ�−iω

18ffiffiffi3

pMðr2 þ 12M2Þ

r3ð6M=r − 1Þ3 þ im72

ffiffiffi2

pM2

r3ð6M=r − 1Þ3

þ 36ffiffiffi6

pM2

r3ð6M=r − 1Þ5=2 −8

ffiffiffi6

pM

r2ð6M=r − 1Þ3=2�exp½iðωtpðrÞ −mφpðrÞÞ�: ð25Þ

We have checked that our results (24) and (25) are inagreement with the corresponding results obtained byHadar and Kol in Ref. [19]. It should be however notedthat we do not use the same conventions for the definitionof the sources, for the Fourier transform as well as the samenormalization for the partial amplitudes ψ ðe=oÞlm ðt; rÞ.Furthermore, it seems to us that our expression for theeven-parity source is much simpler than theirs. It is alsoimportant to note that, due to the relation

Yl−m ¼ ð−1Þm½Ylm��; ð26aÞ

we have

Xl−mφ ¼ ð−1Þm½Xlmφ ��; ð26bÞ

and therefore, as a direct consequence of (26a) and (26b),we can easily observe that

Sðe=oÞωl−m ¼ ð−1Þm½Sðe=oÞ−ωlm��: ð27Þ

The Zerilli-Moncrief and Regge-Wheeler equations (23)can be solved by using the machinery of Green’s functions

(see Ref. [36] for generalities on this topic and, e.g.,Ref. [37] for its use in the context of BH physics). Weconsider the Green’s functions Gðe=oÞωl ðr�; r0�Þ defined by

�d2

dr2�þ ω2 − VlðrÞ

�Gðe=oÞωl ðr�; r0�Þ ¼ −δðr� − r0�Þ ð28Þ

which can be written as

Gðe=oÞωl ðr�;r0�Þ¼−1

Wðe=oÞl ðωÞ

×

8<:ϕ

inðe=oÞωl ðr�Þϕupðe=oÞωl ðr0�Þ; r�r0�:ð29Þ

Here Wðe=oÞl ðωÞ denote the Wronskians of the functionsϕinðe=oÞωl and ϕ

upðe=oÞωl . These functions are linearly indepen-

dent solutions of the homogeneous Zerilli-Moncrief andRegge-Wheeler equations


084008-5

�d2

dr2�þ ω2 − Vðe=oÞl ðrÞ

�ϕðe=oÞωl ¼ 0: ð30Þ

The functions ϕinðe=oÞωl are defined by their purely ingoingbehavior at the event horizon r ¼ 2M (i.e., for r� → −∞)

ϕinðe=oÞωl ðrÞ ∼r�→−∞e−iωr� ð31aÞ

while, at spatial infinity r → þ∞ (i.e., for r� → þ∞), theyhave an asymptotic behavior of the form

ϕinðe=oÞωl ðrÞ ∼r�→þ∞Að−;e=oÞl ðωÞe−iωr� þ Aðþ;e=oÞl ðωÞeþiωr� :

ð31bÞ

Similarly, the functions ϕupðe=oÞωl are defined by their purelyoutgoing behavior at spatial infinity

ϕupðe=oÞðrÞ ∼r�→þ∞

eþiωr� ð32aÞ

and, at the horizon, they have an asymptotic behavior of theform

ϕupðe=oÞωl ðrÞ ∼r�→−∞Bð−;e=oÞl ðωÞe−iωr� þ Bðþ;e=oÞl ðωÞeþiωr� :

ð32bÞ

In the previous expressions, the coefficients Að−;e=oÞl ðωÞ,Aðþ;e=oÞl ðωÞ, Bð−;e=oÞl ðωÞ and Bðþ;e=oÞl ðωÞ are complexamplitudes. By evaluating the Wronskians Wðe=oÞl ðωÞ atr� → −∞ and r� → þ∞, we obtain

Wðe=oÞl ðωÞ ¼ 2iωAð−;e=oÞl ðωÞ ¼ 2iωBðþ;e=oÞl ðωÞ: ð33Þ

Here, it is worth noting some important properties of the

coefficients Að�;e=oÞl ðωÞ and of the functions ϕinðe=oÞωl ðrÞ thatwe will use extensively later. They are a direct consequenceof Eqs. (30) and (31) and they are valid whether ω is real orcomplex. We have

ϕinðe=oÞ−ωl ðrÞ ¼ ½ϕinðe=oÞωl ðrÞ�� ð34aÞ

and

Að�;e=oÞl ð−ωÞ ¼ ½Að�;e=oÞl ðωÞ��: ð34bÞ

It is important to also recall that the solutions of thehomogeneous Zerilli-Moncrief and Regge-Wheeler equa-tions (30) are related by the Chandrasekhar-Detweilertransformation [35,38]

½ΛðΛþ 2Þ − ið12MωÞ�ϕðeÞωl¼

�ΛðΛþ 2Þ þ 72M

2

rðΛrþ 6MÞ fðrÞ þ 12MfðrÞddr

�ϕðoÞωl

ð35Þ

and, as a consequence, the coefficients Að�;e=oÞl ðωÞ satisfythe relations

Að−;eÞl ðωÞ ¼ Að−;oÞl ðωÞ ð36aÞ

and

Aðþ;eÞl ðωÞ ¼ΛðΛþ 2Þ þ ið12MωÞΛðΛþ 2Þ − ið12MωÞ A

ðþ;oÞl ðωÞ: ð36bÞ

Using the Green’s functions (29), we can show that thesolutions of the Moncrief-Zerilli and Regge-Wheeler equa-tions with source (23) are given by

ψ ðe=oÞωlm ðrÞ ¼ −Z þ∞−∞

dr0�Gðe=oÞωl ðr�; r0�ÞSðe=oÞωlm ðr0�Þ ð37aÞ

¼ −Z

6M

2M

dr0

fðr0ÞGðe=oÞωl ðr; r0ÞSðe=oÞωlm ðr0Þ: ð37bÞ

For r → þ∞, the solutions (37b) reduce to the asymp-totic expressions

ψ ðe=oÞωlm ðrÞ ¼eþiωr�

2iωAð−;e=oÞl ðωÞ

Z6M

2M

dr0

fðr0Þϕinðe=oÞωl ðr0ÞSðe=oÞωlm ðr0Þ:

ð38Þ

This result is a consequence of Eqs. (29), (32a) and (33).We can now obtain the solutions of the Zerilli-Moncrief

and Regge-Wheeler equations (10) by inserting (38) into(21) and we have, in the time domain, for the (l, m)waveforms

ψ ðe=oÞlm ðt; rÞ ¼1ffiffiffiffiffiffi2π

pZ þ∞−∞

dω

�e−iω½t−r�ðrÞ�

2iωAð−;e=oÞl ðωÞ

�

×Z

6M

2M

dr0

fðr0Þϕinðe=oÞωl ðr0ÞSðe=oÞωlm ðr0Þ: ð39Þ

Here it is important to note that these partial waveformssatisfy

ψ ðe=oÞl−m ¼ ð−1Þm½ψ ðe=oÞlm ��: ð40Þ

This is a direct consequence of the definition (21) and of therelation


084008-6

ψ ðe=oÞωl−m ¼ ð−1Þm½ψ ðe=oÞ−ωlm�� ð41Þ

which is easily obtained from (38) and (27) by noting that

the solutions ϕinðe=oÞωl of the problem (30)–(31) and theassociated coefficients Að−;e=oÞl ðωÞ satisfy the relations (34).The relations (40) and (26a) permit us to check that thegravitational wave amplitudes (20) are purely real.

III. QUASINORMAL RINGINGS DUE TO THEPLUNGING MASSIVE PARTICLE

In this section, we shall construct the quasinormalringings associated with the gravitational wave amplitudes(20). Of course, they can be obtained by summing overthe ringings associated with all the partial amplitudes

ψ ðe=oÞlm ðt; rÞ. In order to extract from these partial amplitudesthe corresponding quasinormal ringings ψQNMðe=oÞlmn ðt; rÞ, thecontour of integration over ω in Eq. (39) may be deformed(see, e.g., Ref. [39]). This deformation permits us to capturethe zeros of theWronskians (33) lying in the lower part of thecomplexω plane andwhich are the complex frequenciesωlnof the (l, n) QNMs. We note that, for a given l, n ¼ 1corresponds to the fundamental QNM (i.e., the least dampedone) while n ¼ 2; 3;… to the overtones. We also recall thatthe spectrum of the quasinormal frequencies is symmetricwith respect to the imaginary axis, i.e., that if ωln is aquasinormal frequency lying in the fourth quadrant,−ω�ln isthe symmetric quasinormal frequency lying in the third one.We then easily obtain

ψQNMðe=oÞlm ðt; rÞ ¼Xþ∞n¼1

ψQNMðe=oÞlmn ðt; rÞ ð42Þ

with

ψQNMðe=oÞlmn ðt;rÞ¼−

ffiffiffiffiffiffi2π

pðCðe=oÞlmn e−iωln½t−r�ðrÞ�þDðe=oÞlmn eþiω

�ln½t−r�ðrÞ�Þ: ð43Þ

In the previous expression, Cðe=oÞlmn and Dðe=oÞlmn denote the

extrinsic excitation coefficients (see, e.g., Refs. [39–41])which are here defined by

Cðe=oÞlmn ¼Bðe=oÞln�Z

6M

2M

dr0

fðr0Þϕinðe=oÞωl ðr0ÞAðþ;e=oÞl ðωÞ

Sðe=oÞωlm ðr0Þ�ω¼ωln

ð44aÞ

and

Dðe=oÞlmn ¼ ½Bðe=oÞln ��Z

6M

2M

dr0

fðr0Þϕinðe=oÞωl ðr0ÞAðþ;e=oÞl ðωÞ

Sðe=oÞωlm ðr0Þ�ω¼−ω�lnð44bÞ

while

Bðe=oÞln ¼�1

2ω

Aðþ;e=oÞl ðωÞddωA

ð−;e=oÞl ðωÞ

�ω¼ωln

ð45Þ

are the even and odd excitation factors associated with the(l, n) QNM of complex frequency ωln. The first term in theright-hand side (r.h.s.) of Eq. (43) is the contribution ofthe quasinormal frequency ωln lying in the fourth quadrantof the ω plane while the second one is the contributionof −ω�ln, i.e., its symmetric with respect to the imaginaryaxis. In front of the bracket in the r.h.s. of Eq. (44b), the

coefficients ½Bðe=oÞln �� are nothing else than the even and oddexcitation factors associated with the (l, n) QNM ofcomplex frequency −ω�ln. They are obtained from (45) byusing the properties (34b). A few remarks are in order:(1) Thanks to Chandrasekhar and Detweiler, we know

that the zeros of the Wronskians (33), i.e., thequasinormal frequencies ωln (and −ω�ln), do notdepend on the parity sector. This is a consequence ofthe relation (36a).

(2) The excitation factors (45) depend on the paritysector because their expressions involve the coef-ficients Aðþ;e=oÞl ðωÞ which are parity dependent [seeEq. (36b)]. It is interesting to recall that this was notthe case for the problem of the electromagnetic fieldgenerated by charged particle plunging into theSchwarzschild BH [32]. By combining (36) withthe definition (45), we obtain

BðeÞln ¼ΛðΛþ 2Þ þ ið12MωlnÞΛðΛþ 2Þ − ið12MωlnÞ

BðoÞln : ð46Þ

(3) The excitation coefficients (44a) and (44b) dependon the parity sector because they are constructedfrom the excitation factors as well as from waveequations with sources which are parity dependent[see Eq. (23)].

(4) In our problem, the spherical symmetry of theSchwarzschild BH is broken due to the asymmetricplunging trajectory. It is this dissymmetry which, inconnection with the presence of the azimuthal num-ber m, forbids us to gather the two terms in Eq. (43).

(5) It is however important to note that the excitationcoefficients (44) are related by

Cðe=oÞl−mn ¼ ð−1Þm½Dðe=oÞlmn �� ð47Þ

[this is due to the properties (34)] and hence that thequasinormal waveforms (42) satisfy

ψQNMðe=oÞl−m ¼ ð−1Þm½ψQNMðe=oÞlm ��: ð48Þ

The quasinormal gravitational amplitudesobtained from (20) by replacing ψ ðe=oÞlm ðt; rÞ with


084008-7

ψQNMðe=oÞlm ðt; rÞ are then purely real as a consequenceof the relations (48) and (26a).

(6) The ringing amplitudes ψQNMðe=oÞlmn ðt; rÞ andψQNMðe=oÞlm ðt; rÞ do not provide physically relevantresults at “early times” due to their exponentiallydivergent behavior as t decreases. It is necessary todetermine, from physical considerations (see below),the time beyond which these quasinormal wave-forms can be used, i.e., the starting time tstart of theBH ringing.

IV. MULTIPOLAR WAVEFORMS ANDQUASINORMAL RINGDOWNS

A. Numerical methods

In order to construct the gravitational wave amplitudes(20), it is first necessary to obtain numerically the partialamplitudes ψ ðe=oÞlm ðt; rÞ given by (39). For that purpose,using Mathematica [42]:(1) We have determined the functions ϕinðe=oÞωl as well as

the coefficients Að−;e=oÞl ðωÞ. This has been achievedby integrating numerically the homogeneous Zerilli-Moncrief and Regge-Wheeler equations (30) withthe Runge-Kutta method. We have initialized theprocess with Taylor series expansions convergingnear the horizon and we have compared the solutionsto asymptotic expansions with ingoing and outgoingbehavior at spatial infinity that we have decoded byPadé summation. Our numerical calculations havebeen performed independently for the two paritiesand we have checked their robustness and internalconsistency by using the relations (35) and (36).

(2) We have regularized the partial amplitudes ψ ðe=oÞωlm ðrÞgiven by (38), i.e., the Fourier transform of the

partial amplitudes ψ ðe=oÞlm ðt; rÞ. Indeed, these ampli-tudes as integrals over the radial Schwarzschildcoordinate are strongly divergent near the ISCO.This is due to the behavior of the sources (24) and(25) in the limit r → 6M. The regularization processis described in the Appendix. It consists in replacingthe partial amplitudes (38) by their counterparts

(A11) and to evaluate the result by using Levin’salgorithm [33].

(3) We have Fourier transformed ψ ðe=oÞωlm ðrÞ to get thefinal result.

Then, from the partial amplitudes ψ ðe=oÞlm ðt; rÞ, it is possibleto obtain the components hðe=oÞþ and h

ðe=oÞ× of the gravita-

tional signal by using the sums (20).Wehave constructed theeven components from the (l,m) modes with l ¼ 2;…; 10and m ¼ �l which constitute the main contributions.Similarly, we have constructed the odd components fromthe (l, m) modes with l ¼ 2;…; 10 and m ¼ �ðl − 1Þ. Infact, it is not necessary to take higher values forl because, ingeneral, they do not really modify the numerical sums (20)(see also the discussion in Sec. IV B).In order to construct the quasinormal ringings

associated with the gravitational wave amplitudes (20), itis necessary to obtain numerically the partial amplitudes

ψQNMðe=oÞlm ðt; rÞ given by (42) and, as a consequence, weneed the quasinormal frequencies ωln, the excitation

factors Bðe=oÞln as well as the excitation coefficients Cðe=oÞlmn

and Dðe=oÞlmn . For that purpose:(1) We have first determined the quasinormal frequen-

cies ωln by using the method developed by Leaver[43]. We have implemented numerically this methodby using the Hill determinant approach of Majumdarand Panchapakesan [44].

(2) We have then obtained the excitation factors Bðe=oÞln ,as well as the functions ϕinðe=oÞωlnl ðrÞ and the coeffi-cients Aðþ;e=oÞl ðωlnÞ by integrating numerically thehomogeneous Zerilli-Moncrief and Regge-Wheelerequations (30) (for ω ¼ ωln) with the Runge-Kuttamethod and then by comparing the solutions toasymptotic expansions with ingoing and outgoingbehavior at spatial infinity. Our numerical results arein agreement with the theoretical relations (36)and (46).

(3) We have finally determined the excitation coeffi-cients Cðe=oÞlmn and D

ðe=oÞlmn by evaluating the integrals in

Eqs. (44a) and (44b). It should be noted that we donot have to regularize these integrals if we work with

TABLE I. The first quasinormal frequencies ωln and the associated excitation factors Bðe=oÞln .

(l, n) 2Mωln BðeÞln BðoÞln

(2,1) 0.747343 − 0.177925i 0.120928þ 0.070666i 0.126902þ 0.0203151i(3,1) 1.198890 − 0.185406i −0.088969 − 0.061177i −0.093890 − 0.049193i(4,1) 1.618360 − 0.188328i 0.062125þ 0.069099i 0.065348þ 0.065239i(5,1) 2.024590 − 0.189741i −0.036403 − 0.074807i −0.038446 − 0.073524i(6,1) 2.424020 − 0.190532i 0.011847þ 0.075196i 0.013129þ 0.074877i(7,1) 2.819470 − 0.191019i 0.010478 − 0.069864i 0.009689 − 0.069924i(8,1) 3.212390 − 0.191341i −0.029331þ 0.059378i −0.028863þ 0.059573i(9,1) 3.603590 − 0.191565i 0.043628 − 0.044836i 0.043370 − 0.045060i(10,1) 3.993576 − 0.191728i −0.052616þ 0.027669i −0.052494þ 0.027875i


084008-8

weakly damped QNMs. Indeed, let us consider, forexample, the integrals in Eq. (44a) which defines the

excitation coefficients Cðe=oÞlmn . For ω ¼ ωln, due tothe term exp½iωtpðr0Þ�, the sources Sðe=oÞωlm ðr0Þ givenby (24) and (25) vanish exponentially in the limitr0 → 6M and the integrals are convergent at theupper limit. In the limit r0 → 2M, as a consequence of(31a), we have ϕinðe=oÞωl ðr0Þ ∝ ðr0 − 2MÞ−ið2MωÞ and,as a consequence of (5), we have tpðr0Þ≃−2M ln½r0 − 2M� þ Cte. Thus, the integrands areproportional to ðr0 − 2MÞ−ið4MωÞ (see also Sec. IVB of Ref. [19]). By noting thatZ

2Mþϵdr0ðr0 − 2MÞ−ið4MωlnÞ

¼ ϵ−ið4MRe½ωln�Þ

1 − ið4MωlnÞϵ1þð4MIm½ωln�Þ ð49Þ

we can see that the integrals in Eq. (44a) areconvergent at the lower limit 2M if

2MIm½ωln� > −1: ð50Þ

Such a condition is satisfied by the QNMs we shallconsider below.

In fact, for a given l, it is possible to consider only thefundamental QNM (n ¼ 1) which is the least damped one.Moreover, we need only the excitation coefficients CðeÞlmnand DðeÞlmn with l ¼ 2;…; 10 and m ¼ �l and the excita-tion coefficients CðoÞlmn and D

ðoÞlmn with l ¼ 2;…; 10 and

m ¼ �ðl − 1Þ. In Tables I and II, we provide the variousingredients permitting us to construct the quasinormalringings associated with the gravitational wave amplitudes(20). It should be finally recalled that it is necessary toselect a starting time tstart for the ringings. By takingtstart ¼ tpð3MÞ, i.e., the moment the particle crosses thephoton sphere, we have obtained physically relevantresults.

B. Results and comments

In Figs. 2–8, we have considered the components hðe=oÞþ=×of the gravitational waves observed at infinity. The multi-polar waveforms have been obtained by assuming that theparticle starts at r ¼ rISCOð1 − ϵÞ with ϵ ¼ 10−4 and,furthermore, in Eqs. (5) and (6), we have taken φ0 ¼ 0and chosen t0=ð2MÞ in order to shift the interesting part ofthe signal in the window t=ð2MÞ ∈ ½0; 245�. Without loss ofgenerality, we have constructed only the signals fordirections above the orbital plane of the plunging particle.Indeed, we could obtain those observed below that plane byusing the symmetry properties of the vector sphericalharmonics in the antipodal transformation on the unitTA

BLEII.

The

excitatio

ncoefficients

Cðe=oÞ

lmn

andD

ðe=o

Þlmn

correspondingto

thequasinormal

frequenciesωlnandtheexcitatio

nfactorsBðe=o

Þln

ofTableI.

(l,n)

CðeÞ lln

DðeÞ

lln

CðoÞ ll−1n

DðoÞ

ll−1n

(2,1)

−2.2873×10−5−1.1413×10−5i

1.2331×10−7þ1.6133×10−8i

1.3027×10−5−3.7333×10−6i

1.7606×10−8−9.7647×10−7i

(3,1)

−3.4129×10−6þ3.7527×10−7i

−8.9300×10−10−1.6665×10−9i

1.5710×10−6þ2.3923×10−7i

−1.1817×10−8þ4.4988×10−9i

(4,1)

3.4146×10−7−8.3671×10−7i

5.0989×10−11þ2.4768×10−11i

−2.3418×10−7þ3.0764×10−7i

2.4210×10−10−2.4821×10−10i

(5,1)

−2.5796×10−9þ3.2195×10−7i

−1.4574×10−12þ1.4770×10−12i

3.3738×10−8−1.2541×10−7i

8.0274×10−12þ2.2079×10−11i

(6,1)

−2.1803×10−9−1.3600×10−7i

−1.1924×10−13þ5.2103×10−14i

−1.1875×10−8þ5.0871×10−8i

−3.3844×10−12þ5.2475×10−12i

(7,1)

−1.0692×10−8þ6.3204×10−8i

4.0607×10−14−6.8884×10−14i

9.2800×10−9−2.1679×10−8i

−3.0697×10−12þ1.3254×10−12i

(8,1)

1.5158×10−8−2.8867×10−8i

−2.9165×10−14þ3.4046×10−14i

−7.5475×10−9þ8.7255×10−9i

−2.0486×10−12−3.6520×10−13i

(9,1)

−1.3623×10−8þ1.1053×10−8i

2.0391×10−14−2.3094×10−14i

5.3721×10−9−2.6496×10−9i

−1.0116×10−12−9.2597×10−13i

(10,1)

9.6352×10−9−2.0768×10−9i

−1.4482×10−14þ1.2662×10−14i

−3.2557×10−9−1.9158×10−11i

−2.6317×10−13−9.0578×10−13i


084008-9

FIG. 2. Components hðe=oÞþ=× of the gravitational wave observed at infinity in the direction (θ ¼ π=3, φ ¼ 0) (for even components) and(θ ¼ π=6, φ ¼ 0) (for odd components). We emphasize the impact of the harmonics beyond the dominant (l ¼ 2, m ¼ �2) modes onthe multipolar waveforms and their ringdowns (zoom in on the waveforms).


084008-10

2-sphere S2. Moreover, we have assumed that the observerlies in the plane φ ¼ 0. In fact, for any other value of φ, thebehavior of the signals is very similar. The results corre-sponding to arbitrary values of θ and φ are available to theinterested reader upon request.In Figs. 2 and 3, we have focused our attention on the

construction of the multipolar waveforms by summing theexpressions (20) over the harmonics beyond the dominant(l ¼ 2, m ¼ �2) modes. Of course, the necessity to takehigher harmonics into account clearly appears but we canalso note that the sums truncated at l ¼ 5 already providestrong results.The distortion of the multipolar waveforms and of the

associated quasinormal ringdowns clearly appears inFigs. 4–7. It can be observed in the adiabatic phasecorresponding to the quasicircular motion of the particlenear the ISCO (see Fig. 1) as well as in the ringdown phase.It is due to the “large” number of (l, m) modes taken intoaccount in the sums (20) and is strongly dependent on thedirection of the observer.The multipolar waveforms and the associated quasinor-

mal ringdowns are in excellent agreement as can be seenin Figs. 4–7 or, more clearly, in Fig. 8 where we workwith semi-log graphs. Here, it is important to recall (seeSec. IVA) that it has been necessary to regularize theformer while the latter are unregularized.

Finally, in Fig. 9, in order to compare our results withthose obtained in Refs. [19,20], we have displayed themultipolar waveforms and the associated quasinormalringdowns observed at infinity in the orbital plane of theplunging particle, i.e., for θ ¼ π=2. Here, we have onlyconsidered the component hþ ¼ hðeÞþ þ hðoÞþ of the emittedgravitational wave (note that hðeÞ× ¼ hðoÞ× ¼ 0) and we havetaken for the observation directions in the orbital planeφ ¼ 0, φ ¼ π=2, φ ¼ π and φ ¼ 3π=2, i.e., the anglesconsidered by Hadar and Kol in Fig. 4 of Ref. [19]. We canthen realize that the quasinormal ringdowns displayed heredo not agree with those of Hadar and Kol. In fact, we can

recover their results by plotting the sum hðeÞþ − hðoÞþ instead

of hðeÞþ þ hðoÞþ (see Fig. 10 and note that the plots in the rightpanel are in agreement with Fig. 4 of Ref. [19]). Despite acareful study of Ref. [19] and a complete check of our owncalculations, we have not been able to identify the cause ofthis sign difference. Here, it is important to recall thatHadar, Kol, Berti, and Cardoso in Ref. [20] claimed theyhave confirmed the results of Ref. [19] by comparing themwith the Sasaki-Nakamura partial waveforms Xlm. In fact,they have not plotted on a same figure the Sasaki-Nakamura multipolar waveforms and the multipolar qua-sinormal ringdowns. Their comparison is based on anumerical fitting method which is equivalent to compare

FIG. 3. Complement to Fig. 2. Semi-log graphs highlighting the impact on the ringdowns of the harmonics beyond the dominant(l ¼ 2, m ¼ �2) modes.


084008-11

FIG. 4. Multipolar gravitational waveforms hðeÞþ observed at infinity for various directions above the orbital plane of the plungingparticle. We consider φ ¼ 0 and we study the distortion of the multipolar waveform and of the associated quasinormal ringdown when θvaries between −π=2 and þπ=2. We note that, for θ ¼ 0, only the (l ¼ 2, m ¼ �2) modes contribute to the signal.


084008-12

FIG. 5. Multipolar gravitational waveforms hðeÞ× observed at infinity for various directions above the orbital plane of the plungingparticle. We consider φ ¼ 0 and we study the distortion of the multipolar waveform and of the associated quasinormal ringdown when θvaries between −π=2 and þπ=2. We note that hðeÞ× vanishes for θ ¼ �π=2 and that, for θ ¼ 0, only the (l ¼ 2, m ¼ �2) modescontribute to the signal.


084008-13

FIG. 6. Multipolar gravitational waveforms hðoÞþ observed at infinity for various directions above the orbital plane of the plungingparticle. We consider φ ¼ 0 and we study the distortion of the multipolar waveform and of the associated quasinormal ringdown when θvaries between −π=2 and þπ=2. We note that, for θ ¼ 0, only the (l ¼ 3, m ¼ �2) modes contribute to the signal.


084008-14

FIG. 7. Multipolar gravitational waveforms hðoÞ× observed at infinity for various directions above the orbital plane of the plungingparticle. We consider φ ¼ 0 and we study the distortion of the multipolar waveform and of the associated quasinormal ringdown when θvaries between −π=2 and þπ=2. We note that hðoÞ× vanishes for θ ¼ �π=2 and that, for θ ¼ 0, only the (l ¼ 3, m ¼ �2) modescontribute to the signal.


084008-15

FIG. 8. Semi-log graphs of some multipolar waveforms showing the dominance of the quasinormal ringing at intermediate times andthe agreement of the regularized waveforms with the unregularized quasinormal responses.


084008-16

FIG. 9. Multipolar gravitational waveforms hþ ¼ hðeÞþ þ hðoÞþ and associated quasinormal ringdowns observed at infinity for variousdirections in the orbital plane of the plunging particle. The observation directions are φ ¼ 0, φ ¼ π=2, φ ¼ π and φ ¼ 3π=2. The resultsdisplayed in the right panel do not agree with those presented in Fig. 4 of Ref. [19].


084008-17

FIG. 10. Multipolar gravitational waveforms hþ ¼ hðeÞþ − hðoÞþ and associated quasinormal ringdowns observed at infinity for variousdirections in the orbital plane of the plunging particle. The observation directions are φ ¼ 0, φ ¼ π=2, φ ¼ π and φ ¼ 3π=2. The resultsdisplayed in the right panel agree with those presented in Fig. 4 of Ref. [19].


084008-18

the partial modes jXlmj with the quasinormal amplitudesjψQNMðeÞlmn j for lþm even and jψQNMðoÞlmn j for lþm odd. Asa consequence, with this method, a wrong sign in the

combination of hðeÞþ and hðoÞþ cannot be detected.

V. CONCLUSION

In this article,wehave described thegravitational radiationemitted by a massive “point particle” plunging from slightlybelow the ISCO into a Schwarzschild BH. In order to do this,we have constructed the associated multipolar waveformsand analyzed their late-stage ringdown phase in terms ofQNMs. We have noted the excellent agreement between the“exact” waveforms we had to carefully regularize and thecorresponding quasinormal waveforms which have notrequired a similar treatment. Our results have been obtainedfor arbitrary directions of observation and, in particular,outside the orbital plane of the plunging particle. They havepermitted us to emphasize more particularly the impact onthe distortion of the waveforms of (i) the higher harmonicsbeyond the dominant (l ¼ 2, m ¼ �2) modes and (ii) thedirection of observation and, as a consequence, the necessityto take them into account in the analysis of the last phase ofbinary black hole coalescence.

ACKNOWLEDGMENTS

Wegratefully acknowledge Thibault Damour for drawing,some years ago, our attention to the plunge regime ingravitational wave physics and Shahar Hadar and BarakKol for useful correspondence. We wish also to thank YvesDecanini and Julien Queva for various discussions andRomain Franceschini for providing us with powerful com-puting resources.

APPENDIX: REGULARIZATION OF THEPARTIAL WAVEFORM AMPLITUDES

(EVEN AND ODD PARITY)

In this Appendix, we shall explain how to regularize thepartial amplitudes ψ ðe=oÞωlm . Indeed, the exact waveforms (38)as integrals over the radial Schwarzschild coordinate arestrongly divergent near the ISCO. This is due to the behaviorof the sources (24) and (25) in the limit r → 6M. It should benoted that we have encountered a similar problem in ourstudy of the electromagnetic radiation generated by a charged

particle plunging into the Schwarzschild BH [32]. We recallthat we have addressed this problem by combining atheoretical and a numerical approach: for even electromag-netic perturbations, we have reduced the degree of diver-gence of the integrals involved by successive integrations byparts and then we have “numerically regularized” them byusing Levin’s algorithm. Here, we can proceed identicallyand the reader will be assumed to have “in hand” a copy ofRef. [32] where the electromagnetic case is treated in greatdetails in the Appendix. Moreover, we shall use a trick thatwill allow us to quickly provide the expected results fromthose obtained in Ref. [32].The trick we use is based on the fact, that the expressions

(38), which are constructed from the source terms (24) and(25), can be written in the form

ψ ðe=oÞωlm ðrÞ ¼ eiωr�ψ ðe=oÞlm ðωÞ ðA1Þwith

ψ ðe=oÞlm ðωÞ ¼ γðe=oÞZ

6M

2Mdrϕ̃inðe=oÞωl ðrÞÃðe=oÞðrÞeiΦðrÞ ðA2Þ

where we have

ΦðrÞ ¼ ωtpðrÞ −mφpðrÞ ðA3Þand

γðeÞ ¼ 12iωAð−;eÞl ðωÞ

8πm0ffiffiffiffiffiffi2π

p ½Ylmðπ=2; 0Þ��Λþ 2 ; ðA4aÞ

γðoÞ ¼ 12iωAð−;oÞl ðωÞ

16πm0ffiffiffiffiffiffi2π

p ½Xlmφ ðπ=2; 0Þ��ΛðΛþ 2Þ ; ðA4bÞ

and

ϕ̃inðe=oÞωl ¼ κðe=oÞðrÞϕinðe=oÞωl ðrÞ ðA5Þwith

κðeÞðrÞ ¼ − 83

ffiffiffi2

p rΛrþ 6M ; ðA6aÞ

κðoÞðrÞ ¼ffiffiffi3

p �2Mr

�; ðA6bÞ

as well as

ÃðeÞðrÞ ¼ −iω 9rðr2 þ 12M2Þ

ð6M − rÞ3 þ im12

ffiffiffi6

pMr

ð6M − rÞ3 þ18

ffiffiffi2

pM

ffiffiffir

pð6M − rÞ5=2 þ im

�3

ffiffiffi3

p

2ffiffiffi2

p�2Mr2

−ð3 ffiffiffi2p =8Þ ffiffiffirpð6M − rÞ3=2

�3ðΛþ 2Þ − 64M

Λrþ 6M þ72M2ðΛþ 2 −m2Þ

r2þ 216M

3ðΛþ 2 − 2m2ÞΛr3

�; ðA7aÞ

ÃðoÞðrÞ ¼ −iω 9rðr2 þ 12M2Þ

ð6M − rÞ3 þ im12

ffiffiffi6

pMr

ð6M − rÞ3 þ18

ffiffiffi2

pM

ffiffiffir

pð6M − rÞ5=2 −

4ffiffiffi2

p ffiffiffir

pð6M − rÞ3=2 : ðA7bÞ


084008-19

We now remark that the amplitudes ÃðeÞðrÞ and ÃðoÞðrÞcan be split into a divergent and a regular part in the form

Ãðe=oÞðrÞ ¼ ÃdivðrÞ þ Ãðe=oÞreg ðrÞ ðA8Þ

and that the divergent part, which is obtained by the Taylorexpansion of Aðe=oÞðrÞ at r ¼ 6M, is independent of theparity and given by

ÃdivðrÞ ¼c1

ð6M − rÞ3 þc2

ð6M − rÞ5=2 þc3

ð6M − rÞ2 ðA9Þ

with

c1 ¼ 18ið2MÞ2½ffiffiffi6

pm − 36Mω�; ðA10aÞ

c2 ¼ 9ffiffiffi6

pð2MÞ3=2; ðA10bÞ

c3 ¼ 6ið2MÞ½−ffiffiffi6

pmþ 90Mω�: ðA10cÞ

Here, we fall on the result previously obtained in the

context of the regularization of the partial amplitude ψ ðeÞωlmdescribing the electromagnetic radiation generated by acharged particle plunging into the Schwarzschild BH (seeEqs. (A.6)–(A.8) in Ref. [32]). That is a direct consequenceof the redefinition (A5)–(A6) of the functions ϕinðe=oÞωl .Hence, by noting that the functions ϕ̃inðe=oÞωl appearing in(A2) are regular for r → 6M, we can now completethe regularization process by using, mutatis mutandis,the results obtained in Ref. [32] and, in particular,Eq. (A. 21) of this article: In order to regularize the partial

amplitudes ψ ðe=oÞωlm ðrÞ which are given by (38), we thereforereplace in Eq. (A1) the functions ψ ðe=oÞlm ðωÞ by the functions

ψ ðe=oÞreglm ðωÞ¼γðe=oÞZ

6M

2Mdrϕ̃inðe=oÞωl ðrÞÃðe=oÞreg ðrÞeiΦðrÞ

þ3ffiffiffi6

p

2

ffiffiffiffiffiffiffi2M

pγðe=oÞ

�Z6M

2Mdrϕ̃inðe=oÞωl ðrÞ

�1

ð6M−rÞ3=2þ2id

ð6M−rÞ�eiΦðrÞ−2i

Z6M

2MdrrfðrÞϕ̃

inðe=oÞωl ðrÞΘregðrÞð6M−rÞ3=2 e

iΦðrÞ

−2Z

6M

2MdrrfðrÞð

ddrϕ̃

inðe=oÞωl ðrÞÞ

ð6M−rÞ3=2 eiΦðrÞ

�: ðA11Þ

We note that the terms Ãðe=oÞreg in the r.h.s. of (A11) areobtained from (A7)–(A10). Moreover, the function ΘregðrÞis constructed from the phase (A3). This is explained inAppendix of Ref. [32]. We just recall that

ΘregðrÞ ¼ddr

�ΦðrÞ − cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

6M − rp

�−

dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6M − r

p ðA12Þ

where

c ¼ 6ffiffiffiffiffiffiffi2M

pðm − 6

ffiffiffi6

pMωÞ ðA13aÞ

and

d ¼ mþ 12ffiffiffi6

pMω

2ffiffiffiffiffiffiffi2M

p : ðA13bÞ

It is finally important to point out that the result (A11)has to be “numerically regularized.” Indeed (see also thediscussion in Sec. (A.2) of Ref. [32]), the integrands inthe r.h.s. of (A11) belong to a particular family ofrapidly oscillatory functions whose amplitudes diverge as1=ð6M − rÞ3=2 in the limit r → 6M and whose the phaseΦðrÞ behaves as 1=ð6M − rÞ1=2 in the same limit. As aconsequence, it is possible to neutralize the divergencesremaining in the amplitudes from the oscillations inducedby the phase term. This has been achieved by using Levin’salgorithm [33] which is implemented inMathematica [42].It is this last step which permits us to obtain, in Sec. IV,

stable numerical results for the partial amplitudes ψ ðe=oÞωlm ðrÞand ψ ðe=oÞlm ðt; rÞ.


084008-20

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