Black hole (BH) oscillations known as quasi-normal modes (QNMs) are one of
the most important gravitational wave (GW) sources. We propose that higher
perturbative order of QNMs, generated by nonlinear gravitational interaction
near the BHs, are detectable and worth searching for in observations and
simulations of binary BH mergers. We calculate the metric perturbations to
second-order and explicitly regularize the master equation at the horizon and
spatial infinity. We find that the second-order QNMs have frequencies twice the
first-order ones and the GW amplitude is up to ~10% that of the first-order
one. The QNM frequency would also shift blueward up to ~1%. This provides a new
test of general relativity as well as a possible distance indicator.Comment: 5 pages, 1 figure, accepted for publication in PRD Rapid
  Communication

Hiroyuki Nakano

K. D. Kokkotas

Kunihito Ioka

L. D. Landau

English

arXiv

Ioka, K

Nakano, H

Kyoto University Research Information Repository

Title Second- and higher-order quasinormal modes in binary black-hole mergersAuthor(s)Ioka, K; Nakano, HCitationPHYSICAL REVIEW D (2007), 76(6)Issue Date2007-09URL http://hdl.handle.net/2433/49823RightCopyright 2007 American Physical SocietyType Journal ArticleTextversionpublisherKyoto UniversitySecond- and higher-order quasinormal modes in binary black-hole mergersKunihito Ioka1 and Hiroyuki Nakano21Department of Physics, Kyoto University, Kyoto 606-8502, Japan2Center for Computational Relativity and Gravitation, School of Mathematical Sciences, Rochester Institute of Technology,Rochester, New York 14623, USA(Received 25 April 2007; published 13 September 2007)Black-hole (BH) oscillations known as quasinormal modes (QNMs) are one of the most importantgravitational wave (GW) sources. We propose that higher perturbative order of QNMs, generated bynonlinear gravitational interaction near the BHs, are detectable and worth searching for in observationsand simulations of binary BH mergers. We calculate the metric perturbations to second order andexplicitly regularize the master equation at the horizon and spatial infinity. We find that the second-orderQNMs have frequencies twice the first-order ones and the GW amplitude is up to 10% of that of the first-order one. The QNM frequency would also shift blueward up to 1%. This provides a new test of generalrelativity as well as a possible distance indicator.DOI: 10.1103/PhysRevD.76.061503 PACS numbers: 04.30.w, 04.70.Bw, 95.55.Ym, 95.85.SzI. INTRODUCTIONDirect detections of gravitational waves (GWs) willbecome a reality in the near future with current and futuredetectors, such as LIGO, LISA, and DECIGO/BBO [1].GWs can not only provide a test of general relativity butalso open a new window on the universe.One of the most important GW sources are the quasi-normal modes (QNMs) of black holes (BHs) [2]. QNMsare oscillations of the BH metric perturbations, and theyare damped by emitting GWs. QNM frequencies are com-plex with the real part representing the oscillation fre-quency and the imaginary part representing the damping.By observing the QNM frequencies, one can determine themass and angular momentum of spinning BHs.The most promising sources that excite QNMs are bi-nary BH mergers. For these events QNMs will be detectedwith high signal-to-noise ratio (SNR) (e.g., SNR 105 for108M BH mergers at 1 Gpc by LISA) [3] since alarge fraction of the energy (E 1%M for equal-massmergers with a total mass M) goes into GWs. The mergerrate is also estimated to be large enough [4,5]. In numericalrelativity there have been recent breakthroughs for calcu-lating the entire phase of binary BH mergers [6]. The resultis that the ‘  2, m  2 QNM is dominant, carrying away1% of the initial rest mass of the system [7].In this paper we show that the higher-order QNMs arealso prominent in binary BH mergers and they are interest-ing to search for in the observations and simulations. Herethe ‘‘higher-order’’ is with respect to the metric perturba-tions. Although the dimensionless amplitude of the metricperturbations are negligibly small when we observe themat detectors, they are relatively large near the BH, up to 1=M E=M1=2  10% (1)for QNMs with energy E=M 1% [see Eq. (16) with !M1.  1 denotes a wave function to be discussed later].Hence the generated second-order perturbations wouldhave the amplitude 10% of the first-order ones, whichmay be detectable for high SNR events.Higher-order QNMs are essentially analogous to theanharmonic oscillations discussed by Landau andLifshitz in Mechanics [8]. In general, an oscillation withsmall amplitude x is described by an equation, x!2x 0, which gives the first-order solution x  a cos!t.Here a and  are integration constants. Including thesecond-order term with respect to the amplitude x2, theequation becomes x!2x  x2: (2)With the right-hand side as a source term, we obtain asuccessive solution x  a cos!t  x2, where x2  a26!2cos2!t a22!2/ a2: (3)Here an oscillation arises with a frequency 2!. The im-portant point is that the second-order oscillation x2 isdriven by the first-order one and thereby always exists.Our result is that the second-order QNMs would alsohave frequencies twice the first-order ones !2  2!1and amplitude up to 10% of the first-order ones. Sincehigher-order QNMs always exist, we may test the nonline-arity of general relativity. The purpose of this paper is tooutline the calculations of the second-order QNMs for theSchwarzschild BH and clarify the order-of-magnitude es-timates. More details will be given in the forthcomingpapers. We use the units c  G  1, and arbitrarily setM  1 which we can always recover, if necessary.Although this paper is the first to study second-orderQNMs, the second-order analysis is pioneered by Tomita[9], and the ‘  2,m  0 case in the Schwarzschild space-time is studied by Gleiser et al. [10]. It is also extended tocosmology [11] and the Kerr case [12].PHYSICAL REVIEW D 76, 061503(R) (2007)RAPID COMMUNICATIONS1550-7998=2007=76(6)=061503(5) 061503-1 © 2007 The American Physical SocietyII. FIRST ORDERLet us consider the metric perturbations to second order, ~g   g  h1  h2; (4)where g is the Schwarzschild metric and the superscriptsdenote a perturbative order. Expanding the Einstein’s vac-uum equation we can obtain basic equations order by order[10,13].For the first order, we use the Regge-Wheeler-Zerilliformalism [14,15]. Separating angular variables with ten-sor harmonics of indices ‘;m, the equations decouple tothe even (or polar) part with parity 1‘ under a trans-formation ; !  ; and the odd (or axial)part with parity 1‘1. Seven equations for the evenparity part are reduced to a single Zerilli equation, andthe other three equations for the odd parity part to a singleRegge-Wheeler equation. The Zerilli equation is given by  @2@t2 @2@r2	 VZr 1‘mt; r  0;VZr 1 2r22 1r3  62r2  18r 18r3r 32 ;(5)where r	  r 2 lnr=2 1 and   ‘ 1‘ 2=2.By Fourier transforming,  1‘mt; r Rei!t 1‘m!rd!,the Zerilli equation gives a one-dimensional scatteringproblem, which is familiar from quantum mechanics. TheQNMs are obtained by imposing the boundary conditionswith purely ingoing waves  1‘m!  ei!r	 at the horizonand purely outgoing waves  1‘m!  ei!r	 at infinity. Suchboundary conditions are satisfied at discrete QNM frequen-cies !1‘mn that are complex with the imaginary part repre-senting the damping. There are an infinite number ofQNMs for each harmonic ‘;m [16].All physical quantities for the first-order even parity partcan be reconstructed from  1‘m. If we define 1‘m r 1K1‘m r 2Mr 3MH12‘m  r@K1‘m@r(6)in the Regge-Wheeler (RW) gauge according to [17,18],the GW power is given by dE1dt 164X‘m‘ 2!‘ 2!@@t 1‘mt; r2: (7)Note that in the RW gauge the gauge freedom is com-pletely fixed and the physical quantities can be expressedby the gauge invariant functions in simple differentialforms.In this paper we focus on the most dominant modes inbinary BH mergers, i.e., the ‘  2, m  2 even paritymode for the first order [6] and its driving second-ordermode (the ‘  4, m  4 even parity mode as shown be-low). Note that we have to specify m since the degeneracybetween m breaks at the second order.III. SECOND ORDERFor the second order, we also separate angular variablesin terms of tensor harmonics. Instead of  2‘m, we introducea function 2‘m r 2Mr 3Mr2r 2M@K2‘m@tH21‘m(8)in the RW gauge for convenience, which is essentially thetime-derivative of  2‘m [10]. The first-order counterpartexactly satisfies 1‘m  @ 1‘m=@t, and so the dimensionsare  i‘m OM and i‘m OM0 (i  1, 2). The equa-tions for the even parity part are reduced to the Zerilliequation with a source term,  @2@t2 @2@r2	 VZr2‘mt; r  S‘mt; r; (9)where the source term S‘m is quadratic in  1‘m.The most dominant second-order mode is the ‘  4,m  4 even parity one. This is because the dominantfirst-order mode is the ‘  2, m  2 even parity modeand hence S‘m is dominated by the product ‘  2; m 2  ‘  2; m  2 in Eq. (9). This gives them  4 modethat is ‘  4 and even parity. Note that what we areconsidering is the particular solution. Homogeneous solu-tions are just proportional to the first-order ones and aretherefore trivial.IV. REGULARIZATIONAlthough it is straightforward to calculate the ‘  4,m  4 source term S44 in terms of  122 , the raw source termdoes not behave well at infinity and is not suitable forcalculations. We can find S44 Or0 at infinity by usingthe expansion 122 13F00I 1rF0I 1r2FI  F0I Or3; (10)where FI  FIT is some function of T  t r	 andF0I denotes dFIT=dT. We need a regularized sourceSreg44 Or2 at least [i.e., the same as the potential VZ Or2]. At the horizon the source behaves well, S44 Or 2, with 122  F0H 14FH  2756 r 2FH Or 22; (11)where FH  FHT is some function of T  t r	.We can regularize the source term by using the regular-ized function, 2reg44  244 70p126p r 22r@ 122@r@2 122@r@t; (12)KUNIHITO IOKA AND HIROYUKI NAKANO PHYSICAL REVIEW D 76, 061503(R) (2007)RAPID COMMUNICATIONS061503-2which satisfies the Zerilli Eq. (9) with a well-behavedsource term, Sreg44 Or2 at infinity and Sreg44 Or2 at the horizon. Thus we can remove an unphysicalgauge-dependent divergence. Note that such a regulariza-tion is not unique, and, for example, we can replace @=@rwith @=@t in Eq. (12). The regularization is equivalent toadding quadratic terms of the first-order gauge invariantfunction to the second-order gauge invariant function, sothat it preserves the gauge invariance [19].The explicit form of the regularized source term is Sreg44 t; r r 24270pp228r7  8r6  370r5  142r4  384r3  514r2  273r 48r53r 122r 32_ 0 0 72r8  3936r7  2316r6  2030r5  7744r4  9512r3  3540r2  1119r 144r63r 122r 33  0 _  7r 4r3r 22  ::: 24r7  344r6  872r5  771r4  120r3  77r2  237r 48r33r 122r 32r 22 _  66r4  106r3  220r2  156r 45r3r 122r 3r 2 0 _  198r5  318r4  664r3  458r2  127r 243r23r 122r 3r 2  ::: 0 32160r9  11760r8  30560r7  41124r6  31596r5  11630r4  1296r3  4182r2  1341r 144r72r 343r 12  _  7r 43r 0 _ 0  2r 233r 12r2 _ 0  252r6  636r5  674r4  730r3  524r2  171r 24r33r 122r 32r 2   ::: 216r8  4296r7  1992r6  3488r5  8716r4  9512r3  3540r2  1119r 144r63r 122r 33  _ 0; (13)where _  @ 122 =@t and  0  @ 122 =@r. We can nowsolve the regularized Zerilli Eq. (9). With Fourier expan-sions, this provides a two-point boundary value problemwith purely ingoing boundary condition at the horizon andpurely outgoing at infinity. The numerical calculation willbe presented in a forthcoming paper.V. DETECTABILITYWithout solving Eq. (9) numerically, we can find theessential properties of the solutions, i.e., the QNM fre-quency !244n and the order-of-magnitude QNM amplitude.Since the source term S2reg44 is quadratic in the first-orderfunction  122 / ei!122nt, we have 2reg44 / e2i!122nt fromEq. (9). Therefore the second-order QNM frequencies aretwice the first-order ones, !244n  2!122n; (14)which differ from any first-order QNM frequencies !1‘mn.By matching the dimensions in both sides of Eq. (9), wealso estimate the order-of-magnitude amplitude as j2reg44 j  j!122n 122 j2: (15)Then we can derive the second-order QNM energy E2from the first-order one E1. For the QNM waveform, 122   122 0ei!122nt, the first-order energy E1 inEq. (7) is given by E1M 38j!122nj2j 122 0j2Mj=!122nj; (16)where E1=M 1% for equal-mass mergers [6]. By noting244  @ 244 =@t 2!122n 244 , we have a similar expressionfor the second-order energy E2 as E2M 458j2reg44 0j2Mj2=!122njMj=!122njE1M2; (17)where the second equality uses Eqs. (15) and (16). In afuture paper we will calculate the coefficient as E2=M15Mj=!122njE1=M2.Once we know the QNM frequency ! and energy E, wecan obtain the GW energy spectrum [3,20] dEdf’ 162Ef2j=!j3j!j22f<!2  =!22 ; (18)and then the characteristic amplitude hcharfwith Eq. (5.1)of Flanagan and Hughes [3], hcharf2  21 z22Dz2dEdf1 zf; (19)where Dz is the luminosity distance.VI. DISCUSSIONSOur main results are summarized in Fig. 1 showing thecharacteristic GW amplitudes hcharf of the Schwarz-SECOND- AND HIGHER-ORDER QUASINORMAL MODES IN . . . PHYSICAL REVIEW D 76, 061503(R) (2007)RAPID COMMUNICATIONS061503-3schild BH QNMs for two equal-mass binary BH mergers oftotal mass 106M at redshift z  5, together with the rmsnoise amplitude hnf 5fShfpfor the space-baseddetector LISA [3,20] and ultimate DECIGO [1]. Thefirst-order QNM has ‘  2, m  2 and energy E1=M 1%. We estimate the second-order QNM with Eqs. (17)–(19) and the third-order QNM energy by extrapolating thesecond-order Eq. (17) as E3=M 15M=!2E1=M3.We can see that the second- and third-order QNMs appearat 2 and 3 times the first-order frequency, respectively, withdetectable amplitudes.We can in principle identify the higher-order QNMssince their frequencies differ from any first-order ones.However the actual identification depends on the SNR ofthe observations [20] or the accuracy of the simulations. Insimulations with current accuracy we may have alreadymistaken the second-order QNMs for the first-order ones.For example 2M!12m0  0:7473 0:1779i is close toM!14m0  0:80918 0:09416i [16].In order to prove that the second-order QNMs actuallyexist, we have to find the second-order QNMs directly inthe numerical simulations. Such simulations are challeng-ing because the mesh size should be less than 1%M toresolve 1% metric perturbations. Even 1 1D sphericalmodels for the fully self-gravitating case have not foundthe second-order QNMs [21]. Simulations of acousticblack holes may be an alternative for this purpose [22].We also need a mathematically rigorous definition ofsecond-order QNMs like the first-order ones that use theLaplace transformation rather than the Fourier transforma-tion [2,23].Future problems include the Kerr BH case, the oddparity mode case, and a more solid third-order formulation.Since the master equation for Kerr BHs also has a sourceterm quadratic in the first-order function [12], we mayexpect similar results. When BHs have no spin beforemergers, the final spin is a 0:7 [6] and hence the Kerreffects may not be so large (as inferred from the fact thatthe QNM frequencies shift by only a small factor). The oddparity mode appears when BHs have spin before mergers.At the third-order the QNM frequencies will also shift up to 1=M2  1% as suggested by the anharmonic oscilla-tions in [8], probably blueward because the GW carriesaway the BH mass.The ratio between the first- and higher-order QNMamplitudes include new information about the total GWenergy E. Since the observed GW amplitude is hE=M1=2M=r, this could provide a distance indicator.ACKNOWLEDGMENTSWe thank C. O. Lousto and Y. Zlochower for a carefulreading of the manuscript. This work is supported in partby the Grant-in-Aid (18740147) from the Ministry ofEducation, Culture, Sports, Science, and Technology(MEXT) of Japan (K. I.). This is also supported by theJapan Society for the Promotion of Science for ResearchAbroad and in part by the NSF for financial support fromGrant No. PHY-0722315 (H. N.).[1] N. Seto, S. Kawamura, and T. Nakamura, Phys. Rev. Lett.87, 221103 (2001).[2] K. D. Kokkotas and B. G. Schmidt, Living Rev. Relativity2, 2 (1999).[3] E. E. Flanagan and S. A. Hughes, Phys. Rev. D 57, 4535(1998).[4] M. Enoki, K. T. Inoue, M. Nagashima, and N. Sugiyama,Astrophys. J. 615, 19 (2004).FIG. 1. The characteristic GW amplitudes hcharf of theSchwarzschild BH QNMs for two equal-mass binary BH merg-ers of total mass 106M at redshift z  5. The first-order QNMhas ‘  2, m  2 and energy E1=M  1%. The second- andthird-order QNMs also appear at 2 and 3 times the first-orderfrequency, respectively, with detectable amplitudes. The dashedline is the rms noise amplitude hnf 5fShfpfor the space-based detector LISA and ultimate DECIGO. The signal-to-noiseratio squared for a randomly oriented source is given bySNR2  R dlnfhcharf=hnf2. We use a cosmologym;; h  0:27; 0:73; 0:71. We estimate the third-orderQNM energy by extrapolating the second-order Eq. (17) asE3=M 15M=!2E1=M3.KUNIHITO IOKA AND HIROYUKI NAKANO PHYSICAL REVIEW D 76, 061503(R) (2007)RAPID COMMUNICATIONS061503-4[5] K. Ioka and P. Meszaros, Astrophys. J. 635, 143 (2005).[6] F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005); M.Campanelli, C. O. Lousto, P. Marronetti, and Y.Zlochower, Phys. Rev. Lett. 96, 111101 (2006); J. G.Baker, J. Centrella, D. I. Choi, M. Koppitz, and J. vanMeter, Phys. Rev. Lett. 96, 111102 (2006).[7] E. Berti, V. Cardoso, J. A. Gonzalez, U. Sperhake, M.Hannam, S. Husa, and B. Bruegmann, arXiv:gr-qc/0703053.[8] L. D. Landau and E. M. Lifshitz, Mechanics (Butterworth,Heinemann, Oxford, 1976), 3rd ed..[9] K. Tomita, Prog. Theor. Phys. 52, 1188 (1974); K. Tomitaand N. Tajima, Prog. Theor. Phys. 56, 551 (1976).[10] R. J. Gleiser, C. O. Nicasio, R. H. Price, and J. Pullin,Classical Quantum Gravity 13, L117 (1996); Phys. Rev.Lett. 77, 4483 (1996); Phys. Rep. 325, 41 (2000); C. O.Nicasio, R. Gleiser, and J. Pullin, Gen. Relativ. Gravit. 32,2021 (2000).[11] V. Acquaviva, N. Bartolo, S. Matarrese, and A. Riotto,Nucl. Phys. B667, 119 (2003); S. Matarrese, S. Mollerach,and M. Bruni, Phys. Rev. D 58, 043504 (1998); K. Tomita,Phys. Rev. D 71, 083504 (2005); K. Nakamura, Prog.Theor. Phys. 117, 17 (2007).[12] M. Campanelli and C. O. Lousto, Phys. Rev. D 59, 124022(1999).[13] M. Bruni, S. Matarrese, S. Mollerach, and S. Sonego,Classical Quantum Gravity 14, 2585 (1997); K.Nakamura, Prog. Theor. Phys. 110, 723 (2003); D.Brizuela, J. M. Martin-Garcia, and G. A. M. Marugan, J.Phys. Conf. Ser. 66, 012011 (2007).[14] T. Regge and J. Wheeler, Phys. Rev. 108, 1063 (1957).[15] F. J. Zerilli, Phys. Rev. D 2, 2141 (1970).[16] E. W. Leaver, Proc. R. Soc. A 402, 285 (1985).[17] N. Sago, H. Nakano, and M. Sasaki, Phys. Rev. D 67,104017 (2003).[18] K. Martel and E. Poisson, Phys. Rev. D 71, 104003(2005).[19] A. Garat and R. H. Price, Phys. Rev. D 61, 044006(2000).[20] E. Berti, V. Cardoso, and C. M. Will, Phys. Rev. D 73,064030 (2006).[21] Y. Zlochower, R. Gomez, S. Husa, L. Lehner, and J.Winicour, Phys. Rev. D 68, 084014 (2003).[22] S. Okuzumi and M. a. Sakagami, arXiv:gr-qc/0703070.[23] E. W. Leaver, Phys. Rev. D 34, 384 (1986).SECOND- AND HIGHER-ORDER QUASINORMAL MODES IN . . . PHYSICAL REVIEW D 76, 061503(R) (2007)RAPID COMMUNICATIONS061503-5

Physical Review D

Second- and higher-order quasinormal modes in binary black-hole mergers

Black hole (BH) oscillations known as quasi-normal modes (QNMs) are one of the most important gravitational wave (GW) sources. We propose that higher perturbative order of QNMs, generated by nonlinear gravitational interaction near the BHs, are detectable and worth searching for in observations and simulations of binary BH mergers. We calculate the metric perturbations to second-order and explicitly regularize the master equation at the horizon and spatial infinity. We find that the second-order QNMs have frequencies twice the first-order ones and the GW amplitude is up to ∼ 10% that of the first-order one. The QNM frequency would also shift blueward up to ∼ 1%. This provides a new test of general relativity as well as a possible distance indicator

Ioka, Kunihito

Nakano, Hiroyuki

RIT Digital Institutional Repository

Rochester Institute of Technology RIT Digital Institutional Repository Articles Faculty & Staff Scholarship 9-13-2007 Second and Higher-Order Quasinormal Modes in Binary Black Hole Mergers Kunihito Ioka Kyoto University Hiroyuki Nakano Rochester Institute of Technology Follow this and additional works at: https://repository.rit.edu/article Recommended Citation K. Ioka, H. Nakano, Phys. Rev. D 76 061503(R) (2007) https://doi.org/10.1103/PhysRevD.76.061503 This Article is brought to you for free and open access by the RIT Libraries. For more information, please contact repository@rit.edu. arXiv:0704.3467v3  [astro-ph]  3 Aug 2007Second and higher-order quasi-normal modes in binary black hole mergersKunihito Ioka1 and Hiroyuki Nakano21 Department of Physics, Kyoto University, Kyoto 606-8502, Japan2Center for Computational Relativity and Gravitation, School of Mathematical Sciences,Rochester Institute of Technology, Rochester, New York 14623, USABlack hole (BH) oscillations known as quasi-normal modes (QNMs) are one of the most importantgravitational wave (GW) sources. We propose that higher perturbative order of QNMs, generatedby nonlinear gravitational interaction near the BHs, are detectable and worth searching for inobservations and simulations of binary BH mergers. We calculate the metric perturbations tosecond-order and explicitly regularize the master equation at the horizon and spatial infinity. Wefind that the second-order QNMs have frequencies twice the first-order ones and the GW amplitudeis up to ∼ 10% that of the first-order one. The QNM frequency would also shift blueward up to∼ 1%. This provides a new test of general relativity as well as a possible distance indicator.I. INTRODUCTIONDirect detections of gravitational waves (GWs) will be-come a reality in the near future with current and futuredetectors, such as LIGO, LISA and DECIGO/BBO [1].GWs can not only provide a test of general relativity butalso open a new window on the universe.One of the most important GW sources are the quasi-normal modes (QNMs) of black holes (BHs) [2]. QNMsare oscillations of the BH metric perturbations, and theyare damped by emitting GWs. QNM frequencies arecomplex with the real part representing the oscillationfrequency and the imaginary part representing the damp-ing. By observing the QNM frequencies, one can deter-mine the mass and angular momentum of spinning BHs.The most promising sources that excite QNMs are bi-nary BH mergers. For these events QNMs will be de-tected with high signal-to-noise ratio (SNR) (e.g., SNR∼ 105 for ∼ 108M⊙ BH mergers at ∼ 1Gpc by LISA) [3]since a large fraction of energy (E ∼ 1% ×M for equalmass mergers with a total mass M) goes into GWs. Themerger rate is also estimated to be large enough [4, 5].In numerical relativity there has been recently break-throughs for calculating the entire phase of binary BHmergers [6]. The result is that the ℓ = 2, m = 2 QNM isdominant, carrying away ∼ 1% of the initial rest mass ofthe system [7].In this paper we show that the higher-order QNMsare also prominent in binary BH mergers and they areinteresting to search for in the observations and simu-lations. Here the “higher-order” is with respect to themetric perturbations. Although the dimensionless am-plitude of the metric perturbations are negligible smallwhen we observe them at detectors, they are relativelylarge near the BH, up toψ(1)/M ∼ (E/M)1/2 ∼ 10% (1)for QNMs with energy E/M ∼ 1% [see Eq. (16) withω ∼ M−1. ψ(1) denotes a wave function to be discussedlater]. Hence the generated second-order perturbationswould have the amplitude ∼ 10% of the first-order ones,which may be detectable for high SNR events.Higher-order QNMs are essentially analogous to theanharmonic oscillations discussed by Landau & Lifshitzin “Mechanics” [8]. In general, an oscillation with smallamplitude x is described by an equation, ẍ + ω2x = 0,which gives the first-order solution x = a cos(ωt + φ).Here a and φ are integration constants. Including thesecond-order term with respect to the amplitude ∼ x2,the equation becomesẍ+ ω2x = −αx2 . (2)With the right hand side as a source term, we obtain asuccessive solution x = a cos(ωt+ φ) + x(2) wherex(2) =αa26ω2cos 2ωt− αa22ω2∝ a2 . (3)Here an oscillation arises with a frequency 2ω. The im-portant point is that the second-order oscillation x(2) isdriven by the first-order one and thereby always exists.Our result is that the second-order QNMs would alsohave frequencies twice the first-order ones ω(2) = 2ω(1)and the amplitude up to ∼ 10% of the first-order ones.Since higher-order QNMs always exist, we may test thenonlinearity of general relativity. The purpose of thispaper is to outline the calculations of the second-orderQNMs for the Schwarzschild BH and clarify the order-of-magnitude estimates. More details will be given in theforthcoming papers. We use the units c = G = 1, andarbitrarily set M = 1 which we can always recover, ifnecessary.Although this paper is the first to study second-order QNMs, the second-order analysis is pioneeredby Tomita [9], and the ℓ = 2, m = 0 case in theSchwarzschild spacetime is studied by Gleiser et al. [10].It is also extended to cosmology [11] and the Kerrcase [12].II. FIRST-ORDERLet us consider the metric perturbations to second-order,g̃µν = gµν + h(1)µν + h(2)µν , (4)2where gµν is the Schwarzschild metric and the super-scripts denote a perturbative order. Expanding the Ein-stein’s vacuum equation we can obtain basic equationsorder by order [10, 13].For the first-order, we use the Regge-Wheeler-Zerilliformalism [14, 15]. Separating angular variables withtensor harmonics of indices (ℓ,m), the equations decoupleto the even (or polar) part with parity (−1)ℓ under atransformation (θ, φ) → (π − θ, π + φ) and the odd (oraxial) part with parity (−1)ℓ+1. Seven equations for theeven parity part are reduced to a single Zerilli equation,and the other three equations for the odd parity part toa single Regge-Wheeler equation. The Zerilli equation isgiven by[− ∂2∂t2+∂2∂r2∗− VZ(r)]ψ(1)ℓm(t, r) = 0 ; (5)VZ(r) =(1 − 2r)2λ2(λ+ 1)r3 + 6λ2r2 + 18λr + 18r3(λr + 3)2,where r∗ = r+2 ln (r/2 − 1) and λ = (ℓ−1)(ℓ+2)/2. ByFourier transforming, ψ(1)ℓm(t, r) =∫e−iωtψ(1)ℓmω(r)dω, theZerilli equation gives a one-dimensional scattering prob-lem, which is familiar from quantum mechanics. TheQNMs are obtained by imposing the boundary condi-tions with purely ingoing waves ψ(1)ℓmω ∼ e−iωr∗ at thehorizon and purely outgoing waves ψ(1)ℓmω ∼ eiωr∗ at in-finity. Such boundary conditions are satisfied at discreteQNM frequencies ω(1)ℓmn that are complex with the imagi-nary part representing the damping. There is an infinitenumber of QNMs for each harmonics (ℓ,m) [16].All physical quantities for the first-order even paritypart can be reconstructed from ψ(1)ℓm. If we defineψ(1)ℓm =rλ+ 1[K(1)ℓm +r − 2Mλr + 3M(H(1)2ℓm − r∂K(1)ℓm∂r)](6)in the Regge-Wheeler (RW) gauge according to [17, 18],the GW power is given bydE(1)dt=164π∑ℓm(ℓ+ 2)!(ℓ− 2)!∣∣∣∣∂∂tψ(1)ℓm(t, r)∣∣∣∣2. (7)Note that in the RW gauge the gauge freedom is com-pletely fixed and the physical quantities can be expressedby the gauge invariant functions in simple differentialforms.In this paper we focus on the most dominant modesin binary BH mergers, i.e., the ℓ = 2,m = 2 even paritymode for the first-order [6] and its driving second-ordermode (the ℓ = 4,m = 4 even parity mode as shownbelow). Note that we have to specify m since the degen-eracy between m breaks at the second-order.III. SECOND-ORDERFor the second-order, we also separate angular vari-ables in terms of tensor harmonics. Instead of ψ(2)ℓm, weintroduce a functionχ(2)ℓm =r − 2Mλr + 3M[r2r − 2M∂K(2)ℓm∂t−H(2)1ℓm](8)in the RW gauge for convenience, which is essentially thetime-derivative of ψ(2)ℓm [10]. The first-order counterpartexactly satisfies χ(1)ℓm = ∂ψ(1)ℓm/∂t, and so the dimensionsare ψ(i)ℓm ∼ O(M) and χ(i)ℓm ∼ O(M0) (i = 1, 2). Theequations for the even parity part are reduced to theZerilli equation with a source term,[− ∂2∂t2+∂2∂r2∗− VZ(r)]χ(2)ℓm(t, r) = Sℓm(t, r) , (9)where the source term Sℓm is quadratic in ψ(1)ℓm.The most dominant second-order mode is the ℓ = 4,m = 4 even parity one. This is because the dominantfirst-order mode is the ℓ = 2, m = 2 even parity modeand hence Sℓm is dominated by the product (ℓ = 2,m =2) × (ℓ = 2,m = 2) in Eq. (9). This gives the m = 4mode that is ℓ = 4 and even parity. Note that what weare considering is the particular solution. Homogeneoussolutions are just proportional to the first-order ones andare therefore trivial.IV. REGULARIZATIONAlthough it is straightforward to calculate the ℓ = 4,m = 4 source term S44 in terms of ψ(1)22 , the raw sourceterm does not behave well at infinity and is not suitablefor calculations. We can find S44 ∼ O(r0) at infinity byusing the expansionψ(1)22 =13F ′′I +1rF ′I +1r2(FI − F ′I) +O(r−3) , (10)where FI = FI(T−) is some function of T− = t− r∗ andF ′I denotes dFI(T−)/dT−. We need a regularized sourceSreg44 ∼ O(r−2) at least [i.e., the same as the potentialVZ ∼ O(r−2)]. At the horizon the source behaves well,S44 ∼ O[(r − 2)], withψ(1)22 = F′H +14FH +2756(r − 2)FH +O[(r − 2)2] , (11)where FH = FH(T+) is some function of T+ = t+ r∗.We can regularize the source term by using the regu-larized function,χ(2)reg44 = χ(2)44 −√70126√π(r − 2)2r∂ψ(1)22∂r∂2ψ(1)22∂r∂t, (12)3which satisfies the Zerilli equation (9) with a well-behaved source term, Sreg44 ∼ O(r−2) at infinity andSreg44 ∼ O[(r − 2)] at the horizon. Thus we can removean unphysical gauge-dependent divergence. Note thatsuch a regularization is not unique, and for example wecan replace ∂/∂r with ∂/∂t in Eq. (12). The regular-ization is equivalent to adding quadratic terms of thefirst-order gauge invariant function to the second-ordergauge invariant function, so that it preserves the gaugeinvariance [19].The explicit form of the regularized source term isSreg44 (t, r) =r − 242√70√π{228r7 + 8r6 − 370r5 + 142r4 − 384r3 − 514r2 − 273r − 48r5 (3r + 1)2 (2r + 3)2ψ̇′ψ′− 72r8 + 3936r7 + 2316r6 − 2030r5 − 7744r4 − 9512r3 − 3540r2 − 1119r− 144r6 (3r + 1)2(2r + 3)3 ψ′ψ̇+(7r + 4) r3 (r − 2)2ψ···ψ̈ +24r7 + 344r6 − 872r5 − 771r4 + 120r3 + 77r2 − 237r− 48r3 (3r + 1)2(2r + 3)2(r − 2)2ψ̈ψ̇− 66r4 − 106r3 − 220r2 − 156r − 45r (3r + 1)2(2r + 3) (r − 2)ψ̈′ψ̇ +198r5 − 318r4 − 664r3 − 458r2 − 127r− 243r2 (3r + 1)2(2r + 3) (r − 2)ψ···ψ′− 3(2160r9 + 11760r8 + 30560r7 + 41124r6 + 31596r5 + 11630r4 − 1296r3 − 4182r2 − 1341r− 144)r7 (2r + 3)4(3r + 1)2 ψψ̇− 7r + 43rψ̈′ψ̇′ − 2 (r − 2)3 (3r + 1)2r2ψ̈ψ̇′ +252r6 + 636r5 + 674r4 + 730r3 + 524r2 + 171r + 24r3 (3r + 1)2(2r + 3)2(r − 2)ψψ···− 216r8 + 4296r7 + 1992r6 − 3488r5 − 8716r4 − 9512r3 − 3540r2 − 1119r− 144r6 (3r + 1)2 (2r + 3)3ψψ̇′}, (13)where ψ̇ = ∂ψ(1)22 /∂t and ψ′ = ∂ψ(1)22 /∂r. We can nowsolve the regularized Zerilli equation (9). With Fourierexpansions, this provides a two-point boundary valueproblem with purely ingoing boundary condition at thehorizon and purely outgoing at infinity. The numericalcalculation will be presented in a forthcoming paper.V. DETECTABILITYWithout solving Eq. (9) numerically, we can find theessential properties of the solutions, i.e., the QNM fre-quency ω(2)44n and the order-of-magnitude QNM ampli-tude. Since the source term S(2)reg44 is quadratic in thefirst-order function ψ(1)22 ∝ e−iω(1)22nt, we have χ(2)reg44 ∝e−2iω(1)22nt from Eq. (9). Therefore the second-order QNMfrequencies are twice the first-order ones,ω(2)44n = 2ω(1)22n , (14)which differ from any first-order QNM frequencies ω(1)ℓmn.By matching the dimensions in both sides of Eq. (9), wealso estimate the order-of-magnitude amplitude as∣∣∣χ(2)reg44∣∣∣∼∣∣∣ω(1)22nψ(1)22∣∣∣2. (15)Then we can derive the second-order QNM energy E(2)from the first-order one E(1). For the QNM waveform,ψ(1)22 = ψ(1)22 (0)e−iω(1)22nt, the first-order energy E(1) inEq. (7) is given byE(1)M=38π∣∣∣ω(1)22n∣∣∣2 ∣∣∣ψ(1)22 (0)∣∣∣2M∣∣∣ℑω(1)22n∣∣∣, (16)where E(1)/M ∼ 1% for equal-mass mergers [6]. By not-ing χ(2)44 ∼ ∂ψ(2)44 /∂t ∼ 2ω(1)22nψ(2)44 , we have a similar ex-pression for the second-order energy E(2) asE(2)M∼ 458π∣∣∣χ(2)reg44 (0)∣∣∣2M∣∣∣2ℑω(1)22n∣∣∣∼M |ℑω(1)22n|(E(1)M)2, (17)where the second equality uses Eqs. (15) and (16). In anext paper we will calculate the coefficient as E(2)/M ∼15M |ℑω(1)22n|(E(1)/M)2.Once we know the QNM frequency ω and energy E,we can obtain the GW energy spectrum [3, 20]dEdf≃ 16π2Ef2|ℑω|3|ω|2 [(2πf − ℜω)2 + (ℑω)2]2, (18)and then the characteristic amplitude hchar(f) withEq. (5.1) of Flanagan and Hughes [3],hchar(f)2 =2(1 + z)2π2D(z)2dEdf[(1 + z)f ] , (19)where D(z) is the luminosity distance.4FIG. 1: The characteristic GW amplitudes hchar(f) of theSchwarzschild black hole (BH) quasi-normal modes (QNMs)for two equal-mass binary BH mergers of total mass 106M⊙at redshift z = 5. The first-order QNM has ℓ = 2, m = 2 andenergy E(1)/M = 1%. The second and third-order QNMsalso appear at twice and three times the first-order frequency,respectively, with detectable amplitudes. The dashed line isthe rms noise amplitude hn(f) =√5fSh(f) for the space-based detector LISA and ultimate DECIGO. The signal-to-noise ratio squared for a randomly oriented source is given by(SNR)2 =∫d(ln f)[hchar(f)/hn(f)]2. We use a cosmology(Ωm, ΩΛ, h) = (0.27, 0.73, 0.71). We estimate the third-orderQNM energy by extrapolating the second-order equation (17)as E(3)/M ∼ 15(Mℑω)2(E(1)/M)3.VI. DISCUSSIONSOur main results are summarized in Fig. 1 show-ing the characteristic GW amplitudes hchar(f) of theSchwarzschild BH QNMs for two equal-mass binary BHmergers of total mass 106M⊙ at redshift z = 5, togetherwith the rms noise amplitude hn(f) =√5fSh(f) forthe space-based detector LISA [3, 20] and ultimate DE-CIGO [1]. The first-order QNM has ℓ = 2,m = 2 andenergy E(1)/M = 1%. We estimate the second-orderQNM with Eqs. (17)–(19) and the third-order QNM en-ergy by extrapolating the second-order equation (17)as E(3)/M ∼ 15(Mℑω)2(E(1)/M)3. We can see thatthe second and third-order QNMs appear at twice andthree times the first-order frequency, respectively, withdetectable amplitudes.We can in principle identify the higher-order QNMssince their frequencies differ from any first-order ones.However the actual identification depends on the SNRof the observations [20] or the accuracy of the simula-tions. In simulations with current accuracy we may havealready mistaken the second-order QNMs for the first-order ones. For example 2Mω(1)2m0 = 0.7473 − 0.1779i isclose to Mω(1)4m0 = 0.80918− 0.09416i [16].In order to proof that the second-order QNMs actu-ally exist, we have to find the second-order QNMs di-rectly in the numerical simulations. Such simulations arechallenging because the mesh size should be less than∼ 1% ×M to resolve ∼ 1% metric perturbations. Even1+1D spherical models for the fully self-gravitating casehave not found the second-order QNMs [21]. Simula-tions of acoustic black holes may be an alternative forthis purpose [22]. We also need a mathematically rigor-ous definition of second-order QNMs like the first-orderones that use the Laplace transformation rather than theFourier transformation [2, 23].Future problems include the Kerr BH case, the oddparity mode case, and a more solid third-order formula-tion. Since the master equation for Kerr BHs also hasa source term quadratic in the first-order function [12],we may expect similar results. When BHs have no spinbefore mergers, the final spin is a ∼ 0.7 [6] and hencethe Kerr effects may not be so large (as inferred from thefact that the QNM frequencies shift by only a small fac-tor). The odd parity mode appears when BHs have spinbefore mergers. At the third-order the QNM frequen-cies will also shift up to (ψ(1)/M)2 ∼ 1% as suggestedby the anharmonic oscillations in [8], probably bluewardbecause the GW carries away the BH mass.The ratio between the first and higher-order QNMamplitude include new information about the total GWenergy E. Since the observed GW amplitude is h ∼(E/M)1/2(M/r), this could provide a distance indicator.AcknowledgmentsWe thank C. O. Lousto and Y. Zlochower for a care-ful reading of the manuscript. This work is supportedin part by the Grant-in-Aid (18740147) from the Min-istry of Education, Culture, Sports, Science and Tech-nology (MEXT) of Japan (KI). This is also supportedby JSPS Postdoctoral Fellowships for Research Abroadand in part by the NSF for financial support from grantPHY-0722315 (HN).[1] N. Seto, S. Kawamura and T. Nakamura, Phys. Rev.Lett. 87, 221103 (2001).[2] K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel. 2,2 (1999).[3] E. E. Flanagan and S. A. Hughes, Phys. Rev. D 57, 4535(1998).5[4] M. Enoki, K. T. Inoue, M. Nagashima and N. Sugiyama,Astrophys. J. 615, 19 (2004).[5] K. Ioka and P. Meszaros, Astrophys. J. 635, 143 (2005).[6] F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005);M. Campanelli, C. O. Lousto, P. Marronetti and Y. Zlo-chower, Phys. Rev. Lett. 96, 111101 (2006); J. G. Baker,J. Centrella, D. I. Choi, M. Koppitz and J. van Meter,Phys. Rev. Lett. 96, 111102 (2006).[7] E. Berti, V. Cardoso, J. A. Gonzalez, U. Sper-hake, M. Hannam, S. Husa and B. Bruegmann,arXiv:gr-qc/0703053.[8] L. D. Landau and E. M. Lifshitz, Mechanics (Butter-worth, Heinemann, Oxford, 1976), 3rd ed.[9] K. Tomita, Prog. Theor. Phys. 52, 1188 (1974);K. Tomita and N. Tajima, Prog. Theor. Phys. 56, 551(1976).[10] R. J. Gleiser, C. O. Nicasio, R. H. Price and J. Pullin,Class. Quant. Grav. 13, L117 (1996); R. J. Gleiser,C. O. Nicasio, R. H. Price and J. Pullin, Phys. Rev. Lett.77, 4483 (1996); R. J. Gleiser, C. O. Nicasio, R. H. Priceand J. Pullin, Phys. Rept. 325, 41 (2000); C. O. Nicasio,R. Gleiser and J. Pullin, Gen. Rel. Grav. 32, 2021 (2000).[11] V. Acquaviva, N. Bartolo, S. Matarrese and A. Riotto,Nucl. Phys. B 667, 119 (2003); S. Matarrese, S. Moller-ach and M. Bruni, Phys. Rev. D 58, 043504 (1998);K. Tomita, Phys. Rev. D 71, 083504 (2005); K. Naka-mura, Prog. Theor. Phys. 117, 17 (2007).[12] M. Campanelli and C. O. Lousto, Phys. Rev. D 59,124022 (1999).[13] M. Bruni, S. Matarrese, S. Mollerach and S. Sonego,Class. Quant. Grav. 14, 2585 (1997); K. Nakamura, Prog.Theor. Phys. 110, 723 (2003); D. Brizuela, J. M. Martin-Garcia and G. A. M. Marugan, arXiv:gr-qc/0703069.[14] T. Regge and J. Wheeler, Phys. Rev. 108, 1063 (1957).[15] F. J. Zerilli, Phys. Rev. D 2, 2141 (1970).[16] E. W. Leaver, Proc. R. Soc. A 402, 285 (1985).[17] N. Sago, H. Nakano and M. Sasaki, Phys. Rev. D 67,104017 (2003).[18] K. Martel and E. Poisson, Phys. Rev. D 71, 104003(2005).[19] A. Garat and R. H. Price, Phys. Rev. D 61, 044006(2000).[20] E. Berti, V. Cardoso and C. M. Will, Phys. Rev. D 73,064030 (2006).[21] Y. Zlochower, R. Gomez, S. Husa, L. Lehner and J. Wini-cour, Phys. Rev. D 68, 084014 (2003).[22] S. Okuzumi and M. a. Sakagami, arXiv:gr-qc/0703070.[23] E. W. Leaver, Phys. Rev. D 34, 384 (1986).

Second and Higher-Order Quasinormal Modes in Binary Black Hole Mergers

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Rochester Institute of TechnologyRIT Scholar WorksArticles9-13-2007Second and higher-order quasi-normal modes inbinary black hole mergersKunihito IokaHiroyuki NakanoFollow this and additional works at: http://scholarworks.rit.edu/articleThis Article is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Articles by an authorizedadministrator of RIT Scholar Works. For more information, please contact ritscholarworks@rit.edu.Recommended CitationPhysical Review D 76 (2007) 5pgs.arXiv:0704.3467v3  [astro-ph]  3 Aug 2007Second and higher-order quasi-normal modes in binary black hole mergersKunihito Ioka1 and Hiroyuki Nakano21 Department of Physics, Kyoto University, Kyoto 606-8502, Japan2Center for Computational Relativity and Gravitation, School of Mathematical Sciences,Rochester Institute of Technology, Rochester, New York 14623, USABlack hole (BH) oscillations known as quasi-normal modes (QNMs) are one of the most importantgravitational wave (GW) sources. We propose that higher perturbative order of QNMs, generatedby nonlinear gravitational interaction near the BHs, are detectable and worth searching for inobservations and simulations of binary BH mergers. We calculate the metric perturbations tosecond-order and explicitly regularize the master equation at the horizon and spatial infinity. Wefind that the second-order QNMs have frequencies twice the first-order ones and the GW amplitudeis up to ∼ 10% that of the first-order one. The QNM frequency would also shift blueward up to∼ 1%. This provides a new test of general relativity as well as a possible distance indicator.I. INTRODUCTIONDirect detections of gravitational waves (GWs) will be-come a reality in the near future with current and futuredetectors, such as LIGO, LISA and DECIGO/BBO [1].GWs can not only provide a test of general relativity butalso open a new window on the universe.One of the most important GW sources are the quasi-normal modes (QNMs) of black holes (BHs) [2]. QNMsare oscillations of the BH metric perturbations, and theyare damped by emitting GWs. QNM frequencies arecomplex with the real part representing the oscillationfrequency and the imaginary part representing the damp-ing. By observing the QNM frequencies, one can deter-mine the mass and angular momentum of spinning BHs.The most promising sources that excite QNMs are bi-nary BH mergers. For these events QNMs will be de-tected with high signal-to-noise ratio (SNR) (e.g., SNR∼ 105 for ∼ 108M⊙ BH mergers at ∼ 1Gpc by LISA) [3]since a large fraction of energy (E ∼ 1% ×M for equalmass mergers with a total mass M) goes into GWs. Themerger rate is also estimated to be large enough [4, 5].In numerical relativity there has been recently break-throughs for calculating the entire phase of binary BHmergers [6]. The result is that the ℓ = 2, m = 2 QNM isdominant, carrying away ∼ 1% of the initial rest mass ofthe system [7].In this paper we show that the higher-order QNMsare also prominent in binary BH mergers and they areinteresting to search for in the observations and simu-lations. Here the “higher-order” is with respect to themetric perturbations. Although the dimensionless am-plitude of the metric perturbations are negligible smallwhen we observe them at detectors, they are relativelylarge near the BH, up toψ(1)/M ∼ (E/M)1/2 ∼ 10% (1)for QNMs with energy E/M ∼ 1% [see Eq. (16) withω ∼ M−1. ψ(1) denotes a wave function to be discussedlater]. Hence the generated second-order perturbationswould have the amplitude ∼ 10% of the first-order ones,which may be detectable for high SNR events.Higher-order QNMs are essentially analogous to theanharmonic oscillations discussed by Landau & Lifshitzin “Mechanics” [8]. In general, an oscillation with smallamplitude x is described by an equation, x¨ + ω2x = 0,which gives the first-order solution x = a cos(ωt + φ).Here a and φ are integration constants. Including thesecond-order term with respect to the amplitude ∼ x2,the equation becomesx¨+ ω2x = −αx2 . (2)With the right hand side as a source term, we obtain asuccessive solution x = a cos(ωt+ φ) + x(2) wherex(2) =αa26ω2cos 2ωt− αa22ω2∝ a2 . (3)Here an oscillation arises with a frequency 2ω. The im-portant point is that the second-order oscillation x(2) isdriven by the first-order one and thereby always exists.Our result is that the second-order QNMs would alsohave frequencies twice the first-order ones ω(2) = 2ω(1)and the amplitude up to ∼ 10% of the first-order ones.Since higher-order QNMs always exist, we may test thenonlinearity of general relativity. The purpose of thispaper is to outline the calculations of the second-orderQNMs for the Schwarzschild BH and clarify the order-of-magnitude estimates. More details will be given in theforthcoming papers. We use the units c = G = 1, andarbitrarily set M = 1 which we can always recover, ifnecessary.Although this paper is the first to study second-order QNMs, the second-order analysis is pioneeredby Tomita [9], and the ℓ = 2, m = 0 case in theSchwarzschild spacetime is studied by Gleiser et al. [10].It is also extended to cosmology [11] and the Kerrcase [12].II. FIRST-ORDERLet us consider the metric perturbations to second-order,g˜µν = gµν + h(1)µν + h(2)µν , (4)2where gµν is the Schwarzschild metric and the super-scripts denote a perturbative order. Expanding the Ein-stein’s vacuum equation we can obtain basic equationsorder by order [10, 13].For the first-order, we use the Regge-Wheeler-Zerilliformalism [14, 15]. Separating angular variables withtensor harmonics of indices (ℓ,m), the equations decoupleto the even (or polar) part with parity (−1)ℓ under atransformation (θ, φ) → (π − θ, π + φ) and the odd (oraxial) part with parity (−1)ℓ+1. Seven equations for theeven parity part are reduced to a single Zerilli equation,and the other three equations for the odd parity part toa single Regge-Wheeler equation. The Zerilli equation isgiven by[− ∂2∂t2+∂2∂r2∗− VZ(r)]ψ(1)ℓm(t, r) = 0 ; (5)VZ(r) =(1− 2r)2λ2(λ+ 1)r3 + 6λ2r2 + 18λr + 18r3(λr + 3)2,where r∗ = r+2 ln (r/2− 1) and λ = (ℓ−1)(ℓ+2)/2. ByFourier transforming, ψ(1)ℓm(t, r) =∫e−iωtψ(1)ℓmω(r)dω, theZerilli equation gives a one-dimensional scattering prob-lem, which is familiar from quantum mechanics. TheQNMs are obtained by imposing the boundary condi-tions with purely ingoing waves ψ(1)ℓmω ∼ e−iωr∗ at thehorizon and purely outgoing waves ψ(1)ℓmω ∼ eiωr∗ at in-finity. Such boundary conditions are satisfied at discreteQNM frequencies ω(1)ℓmn that are complex with the imagi-nary part representing the damping. There is an infinitenumber of QNMs for each harmonics (ℓ,m) [16].All physical quantities for the first-order even paritypart can be reconstructed from ψ(1)ℓm. If we defineψ(1)ℓm =rλ+ 1[K(1)ℓm +r − 2Mλr + 3M(H(1)2ℓm − r∂K(1)ℓm∂r)](6)in the Regge-Wheeler (RW) gauge according to [17, 18],the GW power is given bydE(1)dt=164π∑ℓm(ℓ+ 2)!(ℓ− 2)!∣∣∣∣ ∂∂tψ(1)ℓm(t, r)∣∣∣∣2. (7)Note that in the RW gauge the gauge freedom is com-pletely fixed and the physical quantities can be expressedby the gauge invariant functions in simple differentialforms.In this paper we focus on the most dominant modesin binary BH mergers, i.e., the ℓ = 2,m = 2 even paritymode for the first-order [6] and its driving second-ordermode (the ℓ = 4,m = 4 even parity mode as shownbelow). Note that we have to specify m since the degen-eracy between m breaks at the second-order.III. SECOND-ORDERFor the second-order, we also separate angular vari-ables in terms of tensor harmonics. Instead of ψ(2)ℓm, weintroduce a functionχ(2)ℓm =r − 2Mλr + 3M[r2r − 2M∂K(2)ℓm∂t−H(2)1ℓm](8)in the RW gauge for convenience, which is essentially thetime-derivative of ψ(2)ℓm [10]. The first-order counterpartexactly satisfies χ(1)ℓm = ∂ψ(1)ℓm/∂t, and so the dimensionsare ψ(i)ℓm ∼ O(M) and χ(i)ℓm ∼ O(M0) (i = 1, 2). Theequations for the even parity part are reduced to theZerilli equation with a source term,[− ∂2∂t2+∂2∂r2∗− VZ(r)]χ(2)ℓm(t, r) = Sℓm(t, r) , (9)where the source term Sℓm is quadratic in ψ(1)ℓm.The most dominant second-order mode is the ℓ = 4,m = 4 even parity one. This is because the dominantfirst-order mode is the ℓ = 2, m = 2 even parity modeand hence Sℓm is dominated by the product (ℓ = 2,m =2) × (ℓ = 2,m = 2) in Eq. (9). This gives the m = 4mode that is ℓ = 4 and even parity. Note that what weare considering is the particular solution. Homogeneoussolutions are just proportional to the first-order ones andare therefore trivial.IV. REGULARIZATIONAlthough it is straightforward to calculate the ℓ = 4,m = 4 source term S44 in terms of ψ(1)22 , the raw sourceterm does not behave well at infinity and is not suitablefor calculations. We can find S44 ∼ O(r0) at infinity byusing the expansionψ(1)22 =13F ′′I +1rF ′I +1r2(FI − F ′I) +O(r−3) , (10)where FI = FI(T−) is some function of T− = t− r∗ andF ′I denotes dFI(T−)/dT−. We need a regularized sourceSreg44 ∼ O(r−2) at least [i.e., the same as the potentialVZ ∼ O(r−2)]. At the horizon the source behaves well,S44 ∼ O[(r − 2)], withψ(1)22 = F′H +14FH +2756(r − 2)FH +O[(r − 2)2] , (11)where FH = FH(T+) is some function of T+ = t+ r∗.We can regularize the source term by using the regu-larized function,χ(2)reg44 = χ(2)44 −√70126√π(r − 2)2r∂ψ(1)22∂r∂2ψ(1)22∂r∂t, (12)3which satisfies the Zerilli equation (9) with a well-behaved source term, Sreg44 ∼ O(r−2) at infinity andSreg44 ∼ O[(r − 2)] at the horizon. Thus we can removean unphysical gauge-dependent divergence. Note thatsuch a regularization is not unique, and for example wecan replace ∂/∂r with ∂/∂t in Eq. (12). The regular-ization is equivalent to adding quadratic terms of thefirst-order gauge invariant function to the second-ordergauge invariant function, so that it preserves the gaugeinvariance [19].The explicit form of the regularized source term isSreg44 (t, r) =r − 242√70√π{228r7 + 8r6 − 370r5 + 142r4 − 384r3 − 514r2 − 273r − 48r5 (3r + 1)2 (2r + 3)2ψ˙′ψ′− 72r8 + 3936r7 + 2316r6 − 2030r5 − 7744r4 − 9512r3 − 3540r2 − 1119r− 144r6 (3r + 1)2(2r + 3)3 ψ′ψ˙+(7r + 4) r3 (r − 2)2ψ···ψ¨ +24r7 + 344r6 − 872r5 − 771r4 + 120r3 + 77r2 − 237r− 48r3 (3r + 1)2(2r + 3)2(r − 2)2 ψ¨ψ˙− 66r4 − 106r3 − 220r2 − 156r − 45r (3r + 1)2(2r + 3) (r − 2) ψ¨′ψ˙ +198r5 − 318r4 − 664r3 − 458r2 − 127r− 243r2 (3r + 1)2(2r + 3) (r − 2) ψ···ψ′− 3(2160r9 + 11760r8 + 30560r7 + 41124r6 + 31596r5 + 11630r4 − 1296r3 − 4182r2 − 1341r− 144)r7 (2r + 3)4(3r + 1)2 ψψ˙− 7r + 43rψ¨′ψ˙′ − 2 (r − 2)3 (3r + 1)2r2ψ¨ψ˙′ +252r6 + 636r5 + 674r4 + 730r3 + 524r2 + 171r + 24r3 (3r + 1)2(2r + 3)2(r − 2) ψψ···− 216r8 + 4296r7 + 1992r6 − 3488r5 − 8716r4 − 9512r3 − 3540r2 − 1119r− 144r6 (3r + 1)2 (2r + 3)3ψψ˙′}, (13)where ψ˙ = ∂ψ(1)22 /∂t and ψ′ = ∂ψ(1)22 /∂r. We can nowsolve the regularized Zerilli equation (9). With Fourierexpansions, this provides a two-point boundary valueproblem with purely ingoing boundary condition at thehorizon and purely outgoing at infinity. The numericalcalculation will be presented in a forthcoming paper.V. DETECTABILITYWithout solving Eq. (9) numerically, we can find theessential properties of the solutions, i.e., the QNM fre-quency ω(2)44n and the order-of-magnitude QNM ampli-tude. Since the source term S(2)reg44 is quadratic in thefirst-order function ψ(1)22 ∝ e−iω(1)22nt, we have χ(2)reg44 ∝e−2iω(1)22nt from Eq. (9). Therefore the second-order QNMfrequencies are twice the first-order ones,ω(2)44n = 2ω(1)22n , (14)which differ from any first-order QNM frequencies ω(1)ℓmn.By matching the dimensions in both sides of Eq. (9), wealso estimate the order-of-magnitude amplitude as∣∣∣χ(2)reg44 ∣∣∣ ∼ ∣∣∣ω(1)22nψ(1)22 ∣∣∣2 . (15)Then we can derive the second-order QNM energy E(2)from the first-order one E(1). For the QNM waveform,ψ(1)22 = ψ(1)22 (0)e−iω(1)22nt, the first-order energy E(1) inEq. (7) is given byE(1)M=38π∣∣∣ω(1)22n∣∣∣2 ∣∣∣ψ(1)22 (0)∣∣∣2M∣∣∣ℑω(1)22n∣∣∣ , (16)where E(1)/M ∼ 1% for equal-mass mergers [6]. By not-ing χ(2)44 ∼ ∂ψ(2)44 /∂t ∼ 2ω(1)22nψ(2)44 , we have a similar ex-pression for the second-order energy E(2) asE(2)M∼ 458π∣∣∣χ(2)reg44 (0)∣∣∣2M∣∣∣2ℑω(1)22n∣∣∣ ∼M |ℑω(1)22n|(E(1)M)2, (17)where the second equality uses Eqs. (15) and (16). In anext paper we will calculate the coefficient as E(2)/M ∼15M |ℑω(1)22n|(E(1)/M)2.Once we know the QNM frequency ω and energy E,we can obtain the GW energy spectrum [3, 20]dEdf≃ 16π2Ef2|ℑω|3|ω|2 [(2πf − ℜω)2 + (ℑω)2]2 , (18)and then the characteristic amplitude hchar(f) withEq. (5.1) of Flanagan and Hughes [3],hchar(f)2 =2(1 + z)2π2D(z)2dEdf[(1 + z)f ] , (19)where D(z) is the luminosity distance.4FIG. 1: The characteristic GW amplitudes hchar(f) of theSchwarzschild black hole (BH) quasi-normal modes (QNMs)for two equal-mass binary BH mergers of total mass 106M⊙at redshift z = 5. The first-order QNM has ℓ = 2,m = 2 andenergy E(1)/M = 1%. The second and third-order QNMsalso appear at twice and three times the first-order frequency,respectively, with detectable amplitudes. The dashed line isthe rms noise amplitude hn(f) =√5fSh(f) for the space-based detector LISA and ultimate DECIGO. The signal-to-noise ratio squared for a randomly oriented source is given by(SNR)2 =∫d(ln f)[hchar(f)/hn(f)]2. We use a cosmology(Ωm,ΩΛ, h) = (0.27, 0.73, 0.71). We estimate the third-orderQNM energy by extrapolating the second-order equation (17)as E(3)/M ∼ 15(Mℑω)2(E(1)/M)3.VI. DISCUSSIONSOur main results are summarized in Fig. 1 show-ing the characteristic GW amplitudes hchar(f) of theSchwarzschild BH QNMs for two equal-mass binary BHmergers of total mass 106M⊙ at redshift z = 5, togetherwith the rms noise amplitude hn(f) =√5fSh(f) forthe space-based detector LISA [3, 20] and ultimate DE-CIGO [1]. The first-order QNM has ℓ = 2,m = 2 andenergy E(1)/M = 1%. We estimate the second-orderQNM with Eqs. (17)–(19) and the third-order QNM en-ergy by extrapolating the second-order equation (17)as E(3)/M ∼ 15(Mℑω)2(E(1)/M)3. We can see thatthe second and third-order QNMs appear at twice andthree times the first-order frequency, respectively, withdetectable amplitudes.We can in principle identify the higher-order QNMssince their frequencies differ from any first-order ones.However the actual identification depends on the SNRof the observations [20] or the accuracy of the simula-tions. In simulations with current accuracy we may havealready mistaken the second-order QNMs for the first-order ones. For example 2Mω(1)2m0 = 0.7473− 0.1779i isclose to Mω(1)4m0 = 0.80918− 0.09416i [16].In order to proof that the second-order QNMs actu-ally exist, we have to find the second-order QNMs di-rectly in the numerical simulations. Such simulations arechallenging because the mesh size should be less than∼ 1% ×M to resolve ∼ 1% metric perturbations. Even1+1D spherical models for the fully self-gravitating casehave not found the second-order QNMs [21]. Simula-tions of acoustic black holes may be an alternative forthis purpose [22]. We also need a mathematically rigor-ous definition of second-order QNMs like the first-orderones that use the Laplace transformation rather than theFourier transformation [2, 23].Future problems include the Kerr BH case, the oddparity mode case, and a more solid third-order formula-tion. Since the master equation for Kerr BHs also hasa source term quadratic in the first-order function [12],we may expect similar results. When BHs have no spinbefore mergers, the final spin is a ∼ 0.7 [6] and hencethe Kerr effects may not be so large (as inferred from thefact that the QNM frequencies shift by only a small fac-tor). The odd parity mode appears when BHs have spinbefore mergers. At the third-order the QNM frequen-cies will also shift up to (ψ(1)/M)2 ∼ 1% as suggestedby the anharmonic oscillations in [8], probably bluewardbecause the GW carries away the BH mass.The ratio between the first and higher-order QNMamplitude include new information about the total GWenergy E. Since the observed GW amplitude is h ∼(E/M)1/2(M/r), this could provide a distance indicator.AcknowledgmentsWe thank C. O. Lousto and Y. Zlochower for a care-ful reading of the manuscript. 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Second and higher-order quasi-normal modes in binary black hole mergers

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