RUP-19-17

Parametrized black hole quasinormal ringdown.II. Coupled equations and quadratic corrections for nonrotating black holes

Ryan McManus,1, ∗ Emanuele Berti,1, † Caio F. B. Macedo,2, ‡ Masashi Kimura,3, § Andrea Maselli,4, ¶ and Vitor Cardoso5, 6, ∗∗1Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, US

2Campus Salinópolis, Universidade Federal do Pará, Salinópolis, Pará, 68721-000 Brazil3Department of Physics, Rikkyo University, Tokyo 171-8501, Japan

4Dipartimento di Fisica, “Sapienza” Università di Roma & Sezione INFN Roma1, P.A. Moro 5, 00185, Roma, Italy5CENTRA, Departamento de Física, Instituto Superior Técnico – IST,

Universidade de Lisboa – UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal6Theoretical Physics Department, CERN 1 Esplanade des Particules, Geneva 23, CH-1211, Switzerland

(Dated: September 2, 2019)

Linear perturbations of spherically symmetric spacetimes in general relativity are described by radial waveequations, with potentials that depend on the spin of the perturbing field. In previous work [1] we studied thequasinormal mode spectrum of spacetimes for which the radial potentials are slightly modified from their generalrelativistic form, writing generic small modifications as a power-series expansion in the radial coordinate. Weassumed that the perturbations in the quasinormal frequencies are linear in some perturbative parameter, andthat there is no coupling between the perturbation equations. In general, matter fields and modifications tothe gravitational field equations lead to coupled wave equations. Here we extend our previous analysis in twoimportant ways: we study second-order corrections in the perturbative parameter, and we address the morecomplex (and realistic) case of coupled wave equations. We highlight the special nature of coupling-inducedcorrections when two of the wave equations have degenerate spectra, and we provide a ready-to-use recipe tocompute quasinormal modes. We illustrate the power of our parametrization by applying it to various examples,including dynamical Chern-Simons gravity, Horndeski gravity and an effective field theory-inspired model.

I. INTRODUCTION

There are experimental and conceptual reasons to expectthat general relativity (GR) and the standard model of particlephysics should be modified at some level. Most modificationsof GR involve additional gravitational degrees of freedom andhigher-order curvature corrections [2, 3]. Black holes (BHs)are a promising experimental playground to reveal or con-strain these modifications. For example, light bosonic fieldscan trigger nonperturbative effects in astrophysical BHs viasuperradiance, affecting the spin distribution of astrophysicalBHs and leading to potentially observable gravitational-waveand electromagnetic signatures [4–11].

In the absence of large, smoking-gun effects, we must relyon precision measurements. Astrophysical BHs in GR are re-markably simple, being characterized only by their mass andspin by virtue of the so-called “no-hair theorems” [12–17]. Assuch, they are ideal laboratories for precision measurements:any deviation from this simplicity is a potential hint of newphysics. In particular, the relaxation of BH spacetimes totheir equilibrium configuration in GR is very simple. Con-sider for example two BHs merging to form a single spinningBH, a process of particular relevance for gravitational-waveastronomy. The merger can be highly dynamical and violent.However, according to GR, at late times the remnant must

∗ rmcmanu3@jhu.edu† berti@jhu.edu‡ caiomacedo@ufpa.br§ mkimura@rikkyo.ac.jp¶ andrea.maselli@roma1.infn.it∗∗ vitor.cardoso@ist.utl.pt

be a slightly perturbed Kerr solution, described by only twoparameters: its mass and spin. The relaxation to a Kerr rem-nant is well described by linear perturbation theory. This isknown as the “ringdown” stage, where the gravitational-waveamplitude consists of a superposition of exponentially dampedsinusoids or “quasinormal modes” (QNMs) with characteristicfrequencies and damping times [18, 19]. A superposition ofQNMs accurately describes the merger waveform even beforethe peak of the gravitational-wave emission [20–27]. There-fore, one possible test of GR consists of testing the consistencybetween vacuum, linearized GR predictions and the observedQNM spectrum. This idea is now commonly called “blackhole spectroscopy” [28–31].The general procedure to test a given modification of GR

is to find BH solutions, compute their QNM spectrum, and fi-nally constrain deviations of the spectrum from the GR predic-tions through gravitational-wave observations. The differen-tial equations that must be solved to determine the QNM spec-trum have a relatively simple, “universal” structure. A gen-eral parametrization of Schwarzschild perturbations inducedby scalar, vector and tensor fields shows that linearized per-turbations always lead to wave-like equations [32–35]. How-ever, in general these wave-like equations are coupled. Ver-ifying which theories lead to coupled perturbation equationsis a laborious task, but some known examples in the litera-ture include the low-energy limit of string-motivated theories,such as Einstein-dilaton-Gauss-Bonnet [36–39] and dynamicalChern-Simons (dCS) gravity [40–44]. Coupling also occursin effective field theory (EFT) modifications of GR [45–48].Even if GR is the correct theory of gravity, matter fields can

couple with each other, and the perturbations of these fieldswill in general be coupled. This happens, for instance, inthe Einstein-Maxwell system [49–55] or for axionic fields in

arX

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charged BH backgrounds [10, 56–58].We have recently computed QNM frequencies for scalar,

vector and tensor perturbations of a spherically symmetricspacetime which can be described as small deviations fromthe corresponding GR perturbation equations (see Ref. [1],henceforth Paper I). We wrote deviations in the correspondingradial potentials as a power series in the (inverse) radial coor-dinate. We found that corrections to the QNM frequencies are(to leading order) linear in these perturbations, and we com-puted the coefficients that determine these corrections. Herewe extend these results by calculating quadratic corrections inthe perturbative potentials, as well as the corrections that arisefrom coupling power-law perturbative corrections between thescalar, vector, polar (even-parity) and axial (odd-parity) grav-itational perturbation equations in GR. We still work underthe assumption that the background solution is nonspinning(although, as shown in Paper I, the formalism can be appliedto spinning black holes in the slow-rotation limit [43, 59]) andthat the perturbation equations are separable.

A. Executive summary

Our starting point is a generalized, matrix-valued masterequation for the coupled radial perturbations induced by Nfields Φ = {Φi} (i = 1, . . . , N):

fddr

(f

dΦdr

)+ [ω2 − f V]Φ = 0 . (1)

Here f = 1 − rH/r , rH = 2M is the horizon radius, ω is thecomplex QNM frequency, and V(r) = Vi j(r) is a N ×N matrixof radial potentials.1 The factor of f ensures that the effectivepotential terms vanish at the horizon.

We assume the background spacetime to be asymptoticallyflat, and we parametrize V as a sum of the GR potentials VGR

i j

and small power-law series corrections δVi j :

Vi j = VGRi j + δVi j , (2)

δVi j =1

r2H

∞∑k=0

α(k)i j

( rHr

)k. (3)

Here VGRi j denotes the potentials describing massless spin s =

0, 1, 2 perturbations in GR. Usually, VGRi j = 0 for i , j and

VGRii , 0, since the fields decouple in GR (but see [49, 58]

for counterexamples). The coefficients α(k)i j are independentof r , but they may be functions of ω [1, 42]. For k ≥ 1 thepotential matrix vanishes at spatial infinity, i.e. Vi j(r) → 0as r → ∞. Perturbations with k = 0 tend to a constantvalue, δVi j → α

(0)i j /r2

H . For simplicity, we neglect off-diagonal

1 In principle the coupled perturbation equations may also involve “friction-like” terms of the form fZ∂rΦ, i.e. terms of first order in radial deriva-tives [34]. However the matrix Z can be reabsorbed into V through fieldredefinitions, as shown in Appendix A.

contributions that fall off slower than the GR potentials [cf.Eqs. (6)–(8) below]: α(0)i j = α

(1)i j = 0 for i , j.

The terms δVi j in the master equation (1) will, in general,modify the GR QNM frequencies ω0 by a correction that isperturbatively small in α(k)i j . The key result of this work is thatthe corrected QNM frequencies read

ω ≈ ω0 + α(k)i j di j

(k) + α(k)i j α

′(s)pqdi j

(k)dpq

(s) +12α(k)i j α

(s)pqei jpq(ks) , (4)

where ei jpq(ks) = epqij(sk) , i, j, p, q = 1, . . . , N , k, s = 0, . . . ,∞, andwe use the Einstein summation convention. A prime denotesa derivative with respect to ω, with all α(s)pq and α′

(s)pq evaluated

at ω0. The derivation of Eq. (4) is presented in Appendix B.The values of ω0 for the tensor, vector and scalar pertur-

bations [19, 30] and the coefficients dii(k) [1] are available on-

line [60]. The values of di j

(k) and ei jpq(ks) for these same per-turbations were first computed in this work, and they are alsoavailable online [60]. We stress that di j

(k) and ei jpq(ks) depend onthe unperturbed potentials VGR

i j and on the unperturbed QNMfrequency ω0.Equation (4) allows for the efficient calculation of the QNM

frequencies to quadratic order through simple multiplicationsand additions. While this expression may look complex, itsuse is trivial: the prefactors α(s)pq and their derivatives are, inprinciple, all independent, with their values prescribed by thespecific theory in question (cf. Section VI for examples).

B. Plan of the paper

The plan of the paper is as follows. In Section II we presentthe eigenvalue problem for the QNM frequencies, and webriefly review the numerical method to find them. In Sec-tion III we compute quadratic corrections for uncoupled fields.In Section IV we show that coupling fields whose spectra arenondegenerate leads to quadratic corrections in the QNM fre-quencies. In Section V we show that coupling fields whosespectra are degenerate leads to corrections in the QNM fre-quencies which are linear in the perturbation parameter. Fi-nally, in Section VI we apply the formalism to some specificexamples: dCS gravity [42, 44], Horndeski gravity [34], andan EFT-inspired model [46]. In Section VII we discuss somelimitations of our analysis and directions for future work.

To improve readability, we relegate several technical resultsto the appendices. As already mentioned, in Appendix A wedemonstrate that friction-like terms (containing first deriva-tives of the fields) can be reabsorbed in the definition of thepotentials, and in Appendix B we derive Eq. (4). In AppendixC we look at the case of three fields. There we show that (i)QNM frequency corrections arising from the coupling of twofields are independent of the total number of coupled fieldsup to quadratic order, and (ii) when two uncoupled fields havenondegenerate spectra, a linear coupling between the fieldsgives rise to quadratic QNM frequency corrections. Finally,in Appendix D we prove that when two uncoupled fields have

3

degenerate spectra, a linear coupling between the fields givesrise to linear QNM frequency corrections.

II. BACKGROUND

A. Quasinormal modes in general relativity

Gravitational perturbations of the Schwarzschild geometryin GR can be classified by their behavior under parity.2 Itis common to classify the metric perturbations as odd (oraxial, or Regge-Wheeler) and even (or polar, or Zerilli). Theseare governed by master variables Φ± which obey the masterequations [62, 63]

fddr

(f

dΦ±dr

)+ [ω2 − f V±]Φ± = 0 . (5)

The effective potential for odd perturbations reads

V− =`(` + 1)

r2 − 3rHr3 , (6)

while the effective potential for even perturbations is

V+ =9λr2

Hr + 3λ2rHr2 + λ2(λ + 2)r3 + 9r3H

r3(λr + 3rH )2, (7)

where λ = `2 + ` − 2, and ` is an angular harmonic indexlabeling the tensorial spherical harmonics used to separate theangular dependence of the perturbations. These two poten-tials are, quite remarkably, isospectral [64, 65], and maintain-ing isospectrality under generic perturbations of the potentialsrequires fine tuning [1].

Perturbations of a Schwarzschild background induced byscalar and vector fields are of interest not only in modi-fied gravity theories (that in general introduce additional de-grees of freedom), but also in phenomena that involve cou-pling between gravitational and nongravitational fields. InGR, Schwarzschild perturbations induced by fields φs of spins = 0, 1 are also described by master equations similar toEq. (5), with potentials [19]

Vs =`(` + 1)

r2 + (1 − s2)rHr3 . (8)

Note that V2 = V−, i.e. the s = 2 potential in Eq. (8) corre-sponds to odd gravitational perturbations.

B. Calculation of the quasinormal frequencies

There are many techniques to compute QNM frequen-cies [19, 43, 66]. Since we are striving for generality, herewe follow a direct integration approach [43]. The idea is to

2 It is possible to construct definite parity perturbations even in the Kerrbackground: see e.g. Appendix C of [61].

integrate the radial wave equations from the horizon to infinitygiven an initial guess of the QNM frequency, and to vary thefrequency until Eq. (1) and the relevant boundary conditionsare satisfied. The values of the fields at the horizon must bespecified to perform the integration. The horizon fields forman N-dimensional vector Φ(i)H and we can perform the inte-gration for any basis of N such Φ(i)H =

{Φ(i)j

}, with the final

result independent of the choice of basis. For simplicity, inour integrations we consider a basis such that Φ(i)j = δ

ij , and

we then construct an N × N matrix S from the integration ofthe N fields under these N initial conditions. The eigenvaluesof Eq. (1) are then the complex roots of

S(ω) ≡ det S(ω) = 0 , (9)

which can be found numerically.As we show below, by expanding Eq. (9) with respect to the

small parameters α(k)i j we get semianalytic expressions for thecoefficients di j

(k) and ei jpq(ks) appearing in Eq. (4).

III. DECOUPLED FIELDS: QUADRATIC CORRECTIONS

Let us start for simplicity from the case of a single field, andtherefore a single nonzero α(k)i j = α in the expansion (3). TheQNM frequencies are the roots of Eq. (9), where the matrixS (now a scalar) is a function of both ω and α. As α andα′ ≡ ∂ωα are in principle independent, we can vary α whileholding α′ constant. By expanding S up to second order in αwe get

S(ω, α) = S|α=0 + αdSdα

����α=0+α2

2d2Sdα2

����α=0

. (10)

Let us restrict the expansion of S(ω, α) to those points suchthat Eq. (9) is satisfied. These points will describe a curveω(α)starting from ω = ω0, α = 0. Replacing the total derivativewith respect to α with the directional derivative along ω(α),one finds

0 = S(ω0, 0) + α(∂

∂α+∂ω

∂α

∂

∂ω

)S����α=0

+α2

2

(∂

∂α+∂ω

∂α

∂

∂ω

)2S����α=0

. (11)

Each term in the expansion must vanish identically along thecurve ω(α). Near α = 0, this curve is approximated by theexpansion ω ≈ ω0 + αd + α2e: cf. Eq. (4). By inserting thisexpansion into the linear term of Eq. (11), we find(

∂

∂α+ (d + αe) ∂

∂ω

)S����α=0= 0 . (12)

We can now evaluate this expression at (ω0, 0) and solve for d:

d = − 1∂ωS|(ω0,0)

∂S∂α

����(ω0,0)

. (13)

4

0 10 20 30 40 50-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0 10 20 30 40 50

-0.0010

-0.0005

0.0000

FIG. 1. Real and imaginary parts of e(kk) [Eq. (4)] for axial gravitational perturbations with selected values of ` and 2 ≤ k ≤ 50.

2 4 6 8 10-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0.001

2 4 6 8 10-0.0015

-0.0010

-0.0005

0.0000

0.0005

FIG. 2. Real and imaginary parts of e(ks) [Eq. (4)] for axial gravitational perturbations with ` = 2, showing the dependence on k at fixed s.The leftmost value for each s corresponds to e(ss), and it is bounded by the ` = 2 curves in Fig. 1.

Following the same steps for the quadratic term in Eq. (11) wefind an expression for e:

e = − 1∂ωS|(ω0,0)

(∂2

∂α2 + 2d∂2

∂α∂ω+ d2 ∂2

∂ω2

)S����(ω0,0)

.

(14)By construction, both of these expressions depend on the struc-ture of S at α = 0, but not on the value of α in the expansion(4), as long as α is small. Therefore these results encode devi-ations from the GR spectrum in a theory-independent manner.

We now reinsert the k index labeling specific power-lawcorrections to the potentials. For linear corrections we get

d(k) = −1

∂ωS|(ω0,0)

∂S∂α(k)

����(ω0,0)

, (15)

while quadratic corrections yield

e(ks) = −1

∂ωS|(ω0,0)

(∂2

∂α(k)∂α(s)+ d(k)

∂2

∂α(s)∂ω

+ d(s)∂2

∂α(k)∂ω+ d(k)d(s)

∂2

∂ω2

)S |(ω0,0) . (16)

Axial Scalark rH e(kk) rH e(kk)

2 - 0.00580 + 0.000345 i - 0.00303 - 0.0000263 i3 - 0.000620 - 0.000470 i - 0.000964 0.000211 i4 0.000731 - 0.00116 i - 0.0000341 - 0.000162 i5 0.000991 - 0.00112 i 0.000250 - 0.000351 i10 0.000678 + 0.0000227 i 0.000263 - 0.0000671 i

TABLE I. The quadratic frequency coefficients e(kk) for the de-coupled odd-parity gravitational and scalar field perturbations with` = 2, as defined in (4). The values for e(ks) with k, s = 0, . . . , 10and ` ≤ 5 for scalar, vector, axial gravitational and polar gravitationalperturbations are available online [60].

Note that Eq. (14) and Eq. (16) agree when k = s, as a resultof the definition of e(ks) in the expansion (4).The linear corrections d(k) where found to five significant

figures in Paper I. Here we present numerical results for thecoefficients e(ks) for a single field perturbed by a power lawpotential.

5

In Table I we list rHe(kk) for axial and scalar gravitationalperturbations with ` = 2 and selected values of k. The valuesfor ` = 2, . . . , 5 and k, s = 0, . . . , 10 for the scalar, vector,axial gravitational and polar gravitational cases are availableonline [60]. The large-k behavior of e(kk) for axial perturba-tions is shown in Fig. 1. In Paper I we found that the linearcoefficients d(k) in the large-k limit are well approximated by

d(j) ≈κ

jβsin(γ ln j + ζ) (17)

where (κ, β, γ, ζ) are numerical coefficients. Assuming thesame functional form for e(kk) for axial perturbations with` = 2, the best-fit parameters are β ≈ 1.7 and γ ≈ 2.4, to becompared with β ≈ 0.66 and γ ≈ 1.5 for d(k). At quadraticorder, cross-term corrections e(ks) also contribute. Repre-sentative correction coefficients e(ks) for axial gravitationalperturbations with k ≥ s and ` = 2 are shown in Fig. 2.

These quadratic corrections are necessary when α > 10−5.The real and imaginary parts of e(kk) and d(k) are of the sameorder of magnitude (compare Table I of Paper I with Table I inthis paper). If α(kk)e(kk) < 10−5d(k) the quadratic correctionwould be smaller than the numerical error in the leading term,which is currently available to five significant figures. By thisargument, we expect the n-th corrections to be needed whenα & 10−5/n.

IV. NONDEGENERATE COUPLED FIELDS

Let us now consider the coupling between any two of thescalar, vector, axial gravitational and polar gravitational per-turbations (excluding for the moment couplings between axialand polar gravitational perturbations, which will be the subjectof Section V below). We will show that di j

(k) is zero for i , j(i.e., that leading-order corrections induced by the couplingsare quadratic) and that the number of coupled fields does notchange the values of dii

(k) = d(k) or ei jpq(ks) . We have producedan extensive list of the coefficients ei jpq(ks) for ` = 2, . . . , 5 andk, s = 0, . . . , 10, which is available online [60].The unperturbed QNM spectrum is the union of the spectra

for each unperturbed field. In this section we assume that theseunperturbed spectra are nondegenerate. Corrections aroundthe tensor QNM spectrum will be called tensor-led in the fol-lowing. Similarly, corrections around the scalar (vector) QNMspectra are scalar- (vector-)led, respectively. Spectra withmul-tiple branches, corresponding to different fields, have been ob-served in extreme-mass ratio simulations and nonlinear BHmergers in Einstein-Maxwell theory [67–70], Chern-Simonstheory [71], and scalar Gauss-Bonnet gravity [39, 72].

The argument used in the derivation of Eqs. (13) and (16)can be generalized with the replacement α(k) → α

i j

(k). Theresult is

di j

(k) = −1

∂ωS|(ω0,0)

∂S∂α(k)i j

������(ω0,0)

(18)

and

ei jpq(ks) = −1

∂ωS|(ω0,0)

(∂2

∂α(k)i j ∂α

(s)pq

+ di j

(k)∂2

∂α(s)pq∂ω

+dpq

(s)∂2

∂α(k)i j ∂ω

+ di j

(k)dpq

(s)∂2

∂ω2

)S

������(ω0,0)

. (19)

While these equations look more complex than Eqs. (13) and(16), they are entirely equivalent but expressed in full gen-erality. Moreover, they reduce to the single-field case wheni = j = p = q.

We now discuss some important subtleties in problems in-volving coupled fields.

A. Effect of N coupled fields

One may worry that if we consider the case of three fieldsand only couple two of them, the resulting di j

(k) may differ fromthe case of only two fields. This is because the roots of Eq. (9)will remain the same when the number of fields increases, butthe functional dependence of the determinant around the rootswill not. Therefore, the derivatives in Eqs. (18) and (19) willbe different depending on the number of fields we consider.Fortunately, the structure of Eq. (9) together with the αi j

(k)expansion implies that we only need to consider the correctionsdue to the roots of the 1×1 and 2×2 minors of Swhich containthe field corresponding to the spectra we perturb about. Hencefor eachω0, one 1×1minor and (N−1) 2×2minors contributecorrections at quadratic order. This is shown in Appendix C.As a result, the value of N plays no role in the calculation ofdi j

(k) or ei jpq(ks) , beyond setting the range of the sums over theindices i and j.The 1×1 minor of the matrix S corresponds to the case of an

uncoupled field. Moreover, the 1×1 minor captures all effectsof the additional potentials to quadratic order. Hence, we canmake the identification dii

(k) = d(k) and eiiii(ks) = e(ks), where d(k)and e(ks) are found from the uncoupled case in Section III.The roots of the 2 × 2 minors will give corrections due

to field couplings up to quadratic order. We will now onlyconsider the case of 2 coupled fields and examine the roots ofthese 2 × 2 minors.

B. Coupling between nondegenerate spectra generatesquadratic corrections

Diagonal perturbation terms δVii(r) generate linear andquadratic corrections to the QNM frequencies. On the con-trary, perturbative couplings δVi j(r) only generate quadraticcorrections, as long as the spectra of the unperturbed fields arenondegenerate. This is shown in Appendix C by examiningthe structure of S.While corrections to the QNM frequencies due to the

coupling between two fields with nondegenerate spectra are

6

Axial-Scalar Polar-Scalark rH e1221

(kk) rH e1221(kk)

2 -0.0388 - 0.00196 i -0.0386 -0.00135 i3 -0.0146 + 0.000930 i -0.0155 + 0.00162 i4 -0.00567 - 0.000484 i -0.00644 + 0.00000923 i5 -0.00228 - 0.00116 i -0.00288 -0.000923 i10 0.000457 - 0.000387 i 0.000318 - 0.000545 i

TABLE II. Quadratic correction coefficients e1221(kk) [cf. Eq. (20)]

for scalar-axial gravitational and scalar-polar gravitational couplings.We show tensor-led ` = 2 corrections for a few selected values of k.The coefficients for k, s = 0, . . . , 10 are available online [60].

quadratic in α, the magnitude of these coupling-induced cor-rections when the values of α are specified need not be smallerthan a given linear correction: for example, in dCS gravityα(k)11 = O(ε) and α

(k)12 = O(ε

1/2) (see e.g. [44] and Sec. VI), sothe two corrections are formally of the same order in ε .

Since there are no linear corrections, Eq. (19) simplifies to

ei jpq(ks) = −1

∂ωS

����(ω0,0)

(∂2

∂α(k)i j ∂α

(s)pq

)S

������(ω0,0)

. (20)

Some representative coefficients for scalar-axial gravitationaland scalar-polar gravitational couplings are listed in Table II.

We stress again that these results only hold when the twounperturbed fields are not isospectral. For example, in thederivation of Eqs. (18) and (19) we have made the assumptionthat ∂ωS|(ω0,0) , 0, which does not hold when the union ofthe unperturbed spectra is degenerate at ω0. This assumptionrules out the important case of a coupling between the axialand polar gravitational perturbations, which are known to beisospectral. We now turn to the effect of couplings betweenfields with degenerate spectra.

V. DEGENERATE COUPLED FIELDS

So far we made the assumption that the spectra of the cou-pled system in Eq. (1) are nondegenerate in the unperturbedcase, i.e. when α(k)i j = 0 for all i, j, k. This was used to obtainEqs. (18) and (19), and also in Appendix C. Unfortunatelythis assumption is not valid whenever there is a coupling be-tween the axial and polar gravitational perturbations, becausein GR the corresponding QNM frequencies are well known tobe isospectral [64, 65].

In this section we show that couplings between degeneratespectra yield linear corrections to the QNM frequencies. Fur-thermore, the expansion (4) does not apply in this case. Ifall elements of δV are nonzero, the total first-order correctionto the spectra is not the sum of the corrections found fromeach individual element of δV, but it is rather a nonlinearcombination of these corrections.

k δV (k)−+ δV(k)+−

2 0.0324 - 0.000828 i3 0.0313 -0.00321 i4 -0.000221 - 0.000215 i5 0.0000165 + 0.00000100 i10 0.00242216 + 0.00309852 i

TABLE III. The product δV (k)−+ δV(k)+− defined in Eqs. (D7) and (D8)

for the case of axial-polar couplings for ` = 2. The coefficients forl = 2 and k, s = 0, . . . , 10 are available online [60].

To illustrate the nature of the problem, let us start again witha simple example. Consider the system defined by(

d2

dr2∗+ ω2 − f V0

)φ1 + αZφ2 = 0 , (21)(

d2

dr2∗+ ω2 − f V0

)φ2 + αZφ1 = 0 , (22)

for some potential V0 and coupling Z . The QNM spectra ofφ1 and φ2 are trivially degenerate for α = 0. To see thatcorrections enter at linear order, diagonalize the system withthe transformation

φ1 = (φ+ + φ−)/2 , (23)φ2 = (φ+ − φ−)/2 . (24)

The resulting equations are then(d2

dr2∗+ ω2 − f V0 + αZ

)φ+ = 0 , (25)(

d2

dr2∗+ ω2 − f V0 − αZ

)φ− = 0 . (26)

It is clear that the corrections to the spectra will enter at lin-ear order in α despite the initial perturbations being coupled,whereas the results of the previous section would imply thatthe correction should be quadratic in α.In Appendix D we consider a general potential matrix of the

form (3). Assuming a QNM frequency expansion of the form

ω = ω0 + εω1, (27)

we show that

ω1 =δV++ + δV−− ±

√(δV++ − δV−−)2 + 4δV+−δV−+

2. (28)

The coefficients δV+− (for example) have the expansion

δV+− =∞∑k=0

a(k)+−δV(k)+− . (29)

Each of the factors δV (k)±± and δV (k)±∓ is related to the expec-tation value of the kth term in a power-series expansion ofthe potential perturbations: see Eqs. (D7) and (D8) for theirdefinitions.

7

0.50

0.55

0.60

0.65

0.70

0.75

0.00 0.02 0.04 0.06 0.08 0.10

-0.30

-0.28

-0.26

-0.24

-0.22

-0.20

-0.18

0 0.05 0.1

-0.5

-0.25

0

0.25

1.0

1.1

1.2

1.3

1.4

0.00 0.02 0.04 0.06 0.08 0.10

-0.30

-0.28

-0.26

-0.24

-0.22

-0.20

0 0.05 0.1

-0.2

-0.1

0

FIG. 3. Fundamental axial ` = 2 QNM frequencies of Schwarzschild BHs in dCS gravity as function of γ̄ for the tensor-led (left) and scalar-led(right) modes. Solid lines refer to Eqs. (33) and (34); bullets were computed through a direct integration method. The inset shows the relativedifference between the two calculations for the real (solid black lines) and imaginary (dashed red lines) part of the modes.

One might have hoped that the corrections to the QNMspectra from the coupling of the two degenerate fields wouldallow for an expansion analogous to Eq. (4). Indeed, in theabsence of coupling, the argument of the square root in Eq. (28)becomes a square, and we do recover a linear sum over single-field expectation values. However, in general, the presence ofcouplings makes this relation nonlinear. Therefore we providethe values of the quantities δV (k)−+ δV

(s)+− for k, s = 0, ...10 and

` = 2 . . . 5 online [60]. In Table III we also list a small sampleof values of δV (k)−+ δV

(k)+− for ` = 2. In order to find linear

corrections to the QNM frequencies, these quantities must beplugged into Eqs. (28) and (29). Note that δV (k)++ and δV (k)−−are just the coefficients d(k) for uncoupled (axial or polar)gravitational perturbations.

VI. EXAMPLES

For illustration, we now apply the formalism to computeQNM spectra for some classes of modified theories of grav-ity that are known to lead to coupled perturbation equations.Specifically, we consider two models where the coupling isbetween scalar and tensor modes (dCS gravity [44] and Horn-deski gravity [34]) and a model where the coupling is betweenaxial and polar gravitational perturbations (the EFT inspiredmodel [46] not considered in detail in Paper I), so that thebackground QNM spectra are degenerate.

A. Dynamical Chern-Simons gravity

In dCS gravity, an effective low-energy theory with an ad-ditional scalar degree of freedom [41], nonspinning BHs aredescribed by the Schwarzschild metric. The polar sector of

gravitational perturbations is the same as in GR, whereas axialgravitational perturbations and scalar perturbations lead to acoupled system of the form (1) [42, 71] with the followingpotentials, in the notation of Eqs. (3), (6) and (8):

V11 = V− , (30)

V12 = V21 =1

r2H

12√βr2

H

√π(` + 2)!(` − 2)!

( rHr

)5, (31)

V22 = Vs=0 +1

r2H

144π`(` + 1)βr4

H

( rHr

)8. (32)

The parameter β appearing in the dCS action has dimensions[L]−4 and it sets the strength of the coupling, playing a rolesimilar to the Brans-Dicke parameterωBD. It is useful to intro-duce a small dimensionless coupling parameter γ̄ ≡ β−1/2r−2

Hsuch that the equations decouple in the GR limit γ̄ → 0.We first study how the parameter γ̄ modifies the tensor-led

mode. Using Eq. (4) and reading off the relevant coefficientsfrom the potentials (30)–(32) we find

ω = ω0 + e1221(55)

©­«12γ̄

√π(` + 2)!(` − 2)!

ª®¬2

. (33)

Proceeding similarly for the scalar-led mode, we find

ω = ω0 + 2d(8)144π`(` + 1)γ̄2 + e(88)[144π`(` + 1)γ̄2]2

+ e1221(55)

©­«12γ̄

√π(` + 2)!(` − 2)!

ª®¬2

. (34)

These expressions illustrate the importance of specifying theform of the coupling: since the off-diagonal terms V12 and V21

8

are proportional to β−1/2, coupling-induced corrections endup being of the same order as the corrections due to δV22.

Tensor-led and scalar-led dCS QNM frequencies have beenpreviosuly computed using various methods [71, 73]. In Fig. 3we compare Eqs. (33) and (34) (solid lines) with a numericalQNM calculation based on the direct integration of the per-turbed field equations (bullets) for the fundamental ` = 2mode. The left panel refers to tensor-led modes, while theright panel refers to scalar-led modes. In the inset we showthe relative difference between the quadratic expansions ofEqs. (33) and (34) and the numerical calculation for the real(solid) and imaginary (dashed) parts of the QNM frequencies.

B. Horndeski gravity

The perturbations of scalar and tensor fields on aSchwarzschild background in Horndeski gravity were studiedin [34] . The even-parity perturbation equations can be sepa-rated through field redefinitions, but there is coupling betweeneven gravitational and scalar perturbations [32].

The master equation for the scalar-led modes takes the form

d2φ

dr2∗+

[ω2 − f

(Vs=0 + µ

2 +`(` + 1)

r2 f Γ)]φ = 0 . (35)

The change to the spectrum is determined by an “effectivemass” parameter µ and by a second parameter Γ, built out ofthe background values (denoted here by overbars) of the freefunctions appearing in the Horndeski action [74, 75]:

µ2 =−G2φφ

3G24φ + G2X − 2G3φ

, (36)

Γ =8G4X

3G24φ + G2X − 2G3φ

. (37)

Let ω0 belong to the spectrum of Vs=0. Then the scalar-ledfrequencies can be approximated as

ω ≈ ω0 + d(0)µ2 + [d(2) + rHd(3)]`(` + 1)Γ + 12

e(00)µ4

+12[e(22) + 2rHe(23) + r2

He(33)][`(` + 1)Γ]2

+ [e(02) + rHe(03)]`(` + 1)µ2Γ . (38)

As noted in [34], for the class of Horndeski theories in whichthe GW speed propagation satisfies cT = 1, the Γ factor van-ishes identically. In this case Eq. (35) depends on a single freeparameter, given by the effective mass µ, and the scalar-ledfrequencies simply reduce to

ω ≈ ω0 + d(0)µ2 +12

e(00)µ4 . (39)

We compute the values ofω for the ` = 2 scalar mode in Horn-deski gravity using a direct integration method as a function ofthe effective mass µ. In Fig. 4 we plot numerical results (bul-lets) against the results obtained from Eq. (39) (solid lines).

● ● ● ● ● ● ● ● ● ● ● ●●

●●

●●

●●

●●

●●

●●

●●

●

●

●

●

perturbed solution● direct integration

0.97

0.974

0.978

0.982

Re[ω

r H]

0.05 0.1 0.15 0.2 0.25

10-9

10-8

10-7

10-6

10-5

μrH

Δω/ω

fit

● ● ● ● ● ● ● ● ● ● ● ●●

●●

●●

●●

●●

●●

●●

●●

●●

●

●

0.05 0.1 0.15 0.2 0.25 0.3-0.194

-0.193

-0.192

-0.191

-0.19

-0.189

μrH

Im[ω

r H]

0.05 0.1 0.15 0.2 0.25

10-8

10-7

10-6

10-5

10-4

μrH

Δω/ω

fit

FIG. 4. Real and imaginary part of the ` = 2 mode of the scalar-ledmode in Horndeski gravity. The numerical results obtained throughdirect integration (bullet points) are in excellent agreement withEq. (39) (solid line). The insets in each panel show the relativedeviation between the two calculations.

The two approaches are in excellent agreement, with relativedeviations ∆ω = (ωdir −ωfit)/ωfit . 10−4 for both the real andimaginary parts within the range of masses we consider.

C. An effective field theory model

One of the EFT models considered in [46] couples the axialand polar gravitational perturbations. Thewave equationsmaponto Eq. (1) with

V11 = V+ , (40)V22 = V− , (41)V12 = V21 = εV(r) , (42)

where V− and V+ were defined in Eqs. (6) and (7), V(r) canbe found in Appendix A of [46], and the small dimensionlessparameter ε is inversely related to the UV cutoff scale of theEFT.The spectra of V+ and V− are degenerate when ε = 0, so

the coupling should induce linear corrections, as discussed inSection V. This expectation is verified in Figure 5. There weconsider perturbations about the fundamental QNM frequencywith ` = 2, and we show how the two branches of QNMfrequencies – corresponding to the two possible sign choicesin Eq. (28) – change with ε . Note that the coupled QNMfrequencies computed by direct integration in Fig. 4 are novelresults that were not presented in [46].

9

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

○ ○ ○ ○ ○○

○○

○○

○○

○○

○○

○○

○○

○

0.74

0.76

0.78

0.8

0.82

0.84

Re[ω

r H]

● ● ● ● ●

○○

○○

○

perturbed solution

0.0 0.005 0.01 0.015 0.020.73

0.74

0.75

0.76

ϵ

Re[ω

r H]

●●

●●

●●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

○○

○○

○○

○○

○○

○○

○○ ○ ○ ○ ○ ○ ○ ○

● direct integration

0.0 0.02 0.04 0.06 0.08 0.1

0.14

0.15

0.16

0.17

0.18

0.19

0.2

ϵ

-Im

[ωr H

]

● ● ● ● ● ● ● ● ● ● ●

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

0.0 0.02 0.040.14

0.16

0.18

0.2

ϵ

-Im

[ωr H

]

FIG. 5. Perturbations of the fundamental QNM frequencies with ` =2 for axial and polar gravitational perturbations, coupled accordingto the EFT model of [46]. The bullet points correspond to frequencyvalues obtained through a direct integration method. The insets showa zoom for small ε where the behavior of ω deviate from the lineartrend, and therefore from the linear approximation given by Eq. (28).

Fitting the two branches of QNM frequencies to a seventh-order polynomial in ε gives a linear coefficient (−0.1479 −0.2729i)rH for the solid black branch, and (0.1480 +0.2719i)rH for the dashed red branch. The linear correctionsare nonzero, as they should, because the uncoupled axial andpolar spectra are degenerate.

VII. CONCLUSIONS AND A COMPUTATIONAL RECIPE

Wehave extended the formalismof Paper I to computeQNMfrequencies of coupled fields as long as the perturbations theyinduce are small deviations from the perturbation equationsfor the Schwarzschild geometry in GR. Our main result is aconvenient, ready-to-use recipe to compute QNM frequenciesat quadratic order in the perturbations. We crucially allowfor the possibility of coupling between the master equations.First-order (friction-like) terms in the field derivatives can beaccommodated through field redefinitions (cf. Appendix A).

We have found the expansion of the QNM frequencies foruncoupled wave equations to quadratic order. Perhaps ourmost interesting findings concern the QNM spectra of coupledfields. When the coupling occurs between fields with nonde-generate spectra at zero order in the perturbations, linear-ordercorrections to the QNM frequencies vanish. However, whenthe coupling occurs between fields with degenerate spectra, the

QNM frequency corrections are linear in the perturbations.Our results significantly simplify the task of computing

QNM frequencies in any modified theory of gravity, or anytheory allowing for additional fields. The general recipe forthis calculation can be summarized as follows:

1) Derive the master equations for the perturbation vari-ables in the given theory;

2) Eliminate first-order (friction-like) terms in the fieldderivatives through field redefinitions, as described inAppendix A;

3) Identify the relevant coefficients α(k)i j in the perturbedpotentials δVi j [Eq. (3)] appearing in the general coupledsystem of Eq. (1); if these coefficients are frequency-dependent, compute their frequency derivative α′(s)pq .

4a) If any two unperturbed spectra are nondegenerate, com-pute corrections to the QNM frequencies by simple mul-tiplications and additions using Eq. (4) and the tabulatedvalues of di j

(k) and ei jpq(ks) , which are available online [60].

4b) If any two unperturbed spectra are degenerate, com-pute corrections to the QNM frequencies using Eq. (28)and the tabulated values of δV (k)±± and δV (k)±∓ defined inEqs. (D7) and (D8), which are also available online [60].

In Sec. VI we illustrate this procedure for three classes ofmodified theories of gravity leading to coupled perturbationequations: two models coupling the scalar and tensor modes(dCS gravity [44] and Horndeski gravity [34]) and an EFTmodel coupling the axial and polar gravitational perturba-tions [46], where the backgroundQNM spectra are degenerate.While our expansion is theory-agnostic, we make assump-

tions about the effect of the modified gravity theory: the back-ground should be perturbatively close to the Schwarzschildmetric, and the corrections to the “ordinary” potentials in theGR master equations should be amenable to a power-seriesexpansion in inverse powers of the radial variable.This results by construction in small corrections to the GR

QNM spectra (4). In general, as discussed in Paper I, newnonperturbative frequencies (e.g., quasibound states emergingfrom zero frequency for massive scalars) may appear in thespectrum, and these are not captured by our formalism.The assumption that the background is only perturbatively

different from the Schwarzschild solution is slightly less re-strictive than one might think. For example, in Paper I weshowed that slowly rotating Kerr BHs can be accommodatedwithin the formalism. Recent work on higher-derivative cor-rections to the Kerr geometry [76] and on QNM frequen-cies of rotating solutions for small coupling [53, 54] mayallow us to make progress on the calculation of QNMs inmodified gravity for rotating BH remnants, such as those ob-served by the LIGO/Virgo collaboration [77]. Related at-tempts at parametrizing deviations from the Kerr QNM spec-trum [78, 79] made use of the connection between the stabilityof null geodesics and QNMs [80–82]. Our formalism may

10

help to clarify the conditions under which this “geodesic cor-respondence” applies [83, 84].Acknowledgments. We thank Macarena Lagos, Oliver Tat-tersall and Aaron Zimmerman for useful discussions. E.B.and R.M. are supported by NSF Grant No. PHY-1841464,NSF Grant No. AST-1841358, NSF-XSEDE Grant No. PHY-090003, and NASA ATP Grant No. 17-ATP17-0225. V.C.acknowledges financial support provided under the EuropeanUnion’s H2020 ERC Consolidator Grant “Matter and strong-field gravity: New frontiers in Einstein’s theory” grant agree-ment no. MaGRaTh–646597. This project has received fund-ing from the European Union’s Horizon 2020 research and in-novation programme under theMarie Sklodowska-Curie grantagreement No 690904. A.M. acknowledge support from theAmaldi Research Center funded by the MIUR program “Di-partimento di Eccellenza” (CUP: B81I18001170001). Weacknowledge financial support provided by FCT/Portugalthrough grant PTDC/MAT-APL/30043/2017. We acknowl-edge the SDSC Comet and TACC Stampede2 clusters throughNSF-XSEDE Award Nos. PHY-090003, as well as MareNos-trum and the technical support provided by Barcelona Super-computing Center (AECT-2018-1-0003). The authors wouldlike to acknowledge networking support by the GWverseCOST Action CA16104, “Black holes, gravitational wavesand fundamental physics.” We acknowledge support from theAmaldi ResearchCenter funded by theMIURprogram “Dipar-timento di Eccellenza” (CUP: B81I18001170001). C.F.B.M.thanks the Conselho Nacional de Desenvolvimento Científicoe Tecnológico (CNPq) for financial support, the JohnsHopkinsUniversity for kind hospitality during the preparation of thiswork and theAmerican Physical Societywhich funded the visitthrough the International Research Travel Award Program.The authors thankfully acknowledge the computer resources,technical expertise and assistance provided by CENTRA/IST.Computations were performed at the cluster “Baltasar-Sete-Sóis” and supported by the H2020 ERC Consolidator Grant“Matter and strong field gravity: New frontiers in Einstein’stheory” grant agreement no. MaGRaTh-646597.

Appendix A: Friction-like terms

Themaster wave equations for coupled fields can, in general,contain “friction-like” terms (i.e., terms which are linear inderivatives of the perturbation variable). Ref. [32] lists severalexamples3 of coupled wave equations of this type.

In this appendix we show that a matrix Z of friction-liketerms (proportional to the first radial derivative of the wavefunction) can always be reabsorbed in the potential matrix Vthrough suitable field redefinitions.

3 See e.g. their Eq. (60) for even-parity scalar-tensor perturbations; Eqs. (88)and (89) for odd-parity vector-tensor perturbations; Eqs. (94)–(97) for even-parity vector-tensor perturbations; and Eq. (104) for even-parity massiveProca perturbations [32].

1. Single-field case

For simplicity, let us first consider adding a friction-liketerm to the master equation:

d2

dr2∗+ εZ(r∗)

dφdr∗+ (ω2 − V)φ = 0 , (A1)

for some function Z(r∗) and small parameter ε (note that inthis appendix, and here only, we redefine f V → V to simplifythe notation). A field redefinition

φ = ξe−ε2∫ r∗

Z(t)dt , (A2)

is sufficient to remove the term proportional to the first deriva-tive [85]. The resulting field equation is

d2ξ

dr2∗+

(ω2 − V − 1

2εZ ′ − 1

4ε2Z2

)ξ = 0 , (A3)

where the last term in parentheses (proportional to ε2Z2) canbe ignored in our perturbative framework. Therefore we canuse the formalism of the main text by a suitable redefinition ofthe radial wave function and of the radial potential. We willnow extend this idea to the case of multiple, coupled fields.

2. Coupled case

Let us now add a matrix of friction-like terms to a coupledset of master equations:

d2Φ

dr2∗+ εZ(r∗)

dΦdr∗+ V (r∗)Φ = 0 , (A4)

for some matrix of functions Z(r∗) and small parameter ε . Forsimplicity, here and below we redefine ω2 − V1,2 → V1,2, weuse primes for derivatives with respect to r∗, and we considerthe case of only two fields.We make a field redefinition

Φ =

(1 − ε

2

∫ r∗Z(t)dt

)χ , (A5)

and we multiply the resulting equations on the left by(1 +

ε

2

∫ r∗Z(t)dt

). (A6)

This yields

d2χ

dr2∗+ (V + εW )χ = 0 , (A7)

11

whereW is a matrix with elements

W11 =12

(V21

∫ r∗Z12dr ′∗ − V12

∫ r∗Z21dr ′∗ − Z ′11

), (A8)

W22 =12

(V12

∫ r∗Z21dr ′∗ − V21

∫ r∗Z12dr ′∗ − Z ′22

), (A9)

W12 =12

(V12

∫ r∗(Z11 − Z22)dr ′∗

+ (V22 − V11)∫ r∗

Z12dr ′∗ − Z ′12

), (A10)

W21 =12

(V21

∫ r∗(Z22 − Z11)dr ′∗

+ (V11 − V22)∫ r∗

Z21dr ′∗ − Z ′12

). (A11)

As a result, the field equations no longer depend on the firstderivative of any field. When V and Z are diagonal we recoverEq. (A3) at linear order in ε , and when ε = 0 we recover theunperturbed QNM spectrum.

In general, the removal of friction-like terms introduces newpotentials. For these new potentials to fit into our perturbativeformalism, all new contributions must vanish at the horizonand (atmost) tend to a constant at infinity. One parametrizationthat would satisfy these requirements is

εZi j =1

rH

(1 − rH

r

)2 ∑s≥2

β(s)i j

rsHrs

(A12)

for small parameters β(s)i j , which can be mapped to the param-eters α(k)i j discussed in the body of the paper. The summationmust start at s = 2, so that the integrals in Eqs. (A8)–(A11) donot diverge as r∗ →∞.

Appendix B: Expansion of ω

The coupling parameters α(k)i j appearing in Eq. (3) can, ingeneral, depend on the frequency ω. The computation of theQNM frequency itself depends on the couplings, so a non-trivial ω-dependence of the couplings will affect the QNMs.Consider the implicit expansion of ω about the GR value ω0:

ω = ω0 + α(k)i j (ω)d

i j

(k) +12α(k)i j (ω)α

(s)pq(ω)ei jpq(ks) + O(α

3) .(B1)

One may also Taylor expand αki j(ω) about ω0,

α(k)i j (ω) = α

(k)i j |ω0 + α

′(k)i j |ω0 (ω − ω0) + O(α2) . (B2)

We assume that α(k)i j and all of its derivatives evaluated at ω0are small, so that we can consider them to be of the same order.By substituting (B2) into (B1) we find

ω ≈ ω0 + [α(k)i j |ω0 + α′(k)i j |ω0 (ω − ω0)]di j

(k)

+12α(k)i j |ω0α

(s)pq |ω0 ei jpq(ks) + O(α

3) . (B3)

Finally, substitute (B3) into itself to get the result quoted inEq. (4):

ω ≈ ω0 + α(k)i j |ω0 di j

(k) + α′(k)i j |ω0α

(s)pq |ω0 di j

(k)dpq

(s)

+12α(k)i j |ω0α

(s)pq |ω0 ei jpq(ks) + O(α

3) . (B4)

Appendix C: The case of three fields

Let us schematically write the coupled master equations forthree fields as

©­­«L1 + δV11 δV12 δV13

δV21 L2 + δV22 δV23

δV31 δV32 L3 + δV33

ª®®¬©­­«φ

ψ

χ

ª®®¬ = ω2 ©­­«φ

ψ

χ

ª®®¬ (C1)

for some linear operators L1,2,3, perturbative potentials δVi j =

αi jV̄i j and small parameters αi j (in this appendix we slightlychange the notation to minimize clutter).We assume that the spectrum of L1 is nondegenerate with

the spectra of both L2 and L3. Expanding the fields in powersof αi j to linear order we find

φ = φ0 + α11φ11 + α12φ12 + α13φ13 , (C2)ψ = ψ0 + α21ψ21 + α22ψ22 + α23ψ23 , (C3)χ = χ0 + α31 χ31 + α32 χ32 + α33 χ33 . (C4)

Note that all of the first-order fields are only functions of theuncoupled zeroth-order fields.

Let us write down explicitly some of the field equations:

ω2φ0 = L1φ0 , (C5)ω2φ12 = L1φ12 + V̄12ψ0 , (C6)ω2ψ23 = L1ψ23 + V̄23 χ0 . (C7)

Recall from Section II B that we numerically integrate thesystem of equations order by order with respect to a diagonalbasis of initial values at the horizon. We also need to specifythe value of each perturbation at the horizon, which we set tozero for all but the zeroth-order component of each field, sowhen αi j = 0 the perturbations are not excited. We will denoteby Nψ(φ12), for example, the numerically integrated solutionto Eq. (C6). In this notation, the superscript denotes whichfield is excited at the horizon during the integration.

The purpose of this appendix is to show that the perturbedspectrum of L1 is calculated using Eqs. (18) and (19), wherethe determinant S refers to the 1 × 1 or 2 × 2 minors of the3 × 3 matrix

S =©­­«

Nφ(φ) Nφ(ψ) Nφ(χ)Nψ(φ) Nψ(ψ) Nψ(χ)Nχ(φ) Nχ(ψ) Nχ(χ)

ª®®¬ (C8)

which contain Nφ(φ). The proof goes as follows.Let ω be in the spectrum of φ0, so that Nφ(φ0) = 0: cf.

Eq. (C5). Then we can conclude that:

12

(i) Nφ(ψ0) = Nφ(χ0) = 0, Nψ(φ0) = Nψ(χ0) = 0 andNχ(φ0) = Nχ(ψ0) = 0, as the leading order is not ex-cited by our choice of basis.

(ii) Nφ(φ12) = Nφ(φ13) = 0: this is because the equationsfor φ12 and φ13 are sourced by ψ0 and χ0 [cf. Eq. (C6)],which are zero by our choice of basis.

(iii) Nφ(ψ23) = 0, and similar relations apply under permu-tations φ → ψ → χ. This is because the equations forψ23 is sourced by χ0 [cf. Eq. (C7)], which is zero by ourchoice of basis.

In conclusion, the matrix S reduces to

©­­«α11Nφ(φ11) α21Nφ(ψ21) α31Nφ(χ31)α12Nψ(φ12) Nψ(ψ0) + α22Nψ(ψ22) α32Nψ(χ32)α13Nχ(φ13) α23Nχ(ψ23) Nχ(χ0) + α33Nχ(χ33)

ª®®¬ ,(C9)

and its determinant readsS = α11Nφ(φ11)Nψ(ψ0)Nχ(χ0)− α12α21Nφ(ψ21)Nψ(φ12)Nχ(χ0)− α13α31Nχ(φ13)Nφ(χ31)Nψ(ψ0)+ α11α33Nφ(φ11)Nχ(χ33)Nψ(ψ0)− α11α22Nφ(φ11)Nψ(ψ22)Nχ(χ0)+ O(α3) = 0 . (C10)

The αi j coefficients of each term can be varied indepen-dently, so each term in this sum must vanish. Note also thatNψ(ψ0) , 0 and Nχ(χ0) , 0, as we have assumed that L1 isnot degenerate with L2 and L3. The first three lines in thissum give the corrections to linear and quadratic order, and areexplicitly those formed from the 1×1 and the 2×2 minors of Scontaining the (1, 1) element, because we perturb around thespectrum of L1. The last two lines vanish identically, because– recalling that Nψ(ψ0) , 0 and Nχ(χ0) , 0 – the first lineimplies Nφ(φ11) = 0.In conclusion: adding a third field does not affect our ex-

pansion, and the roots of the 1×1 or 2×2 minors of the matrixS [Eq. (C8)] which contain Nφ(φ) can be used to compute theappropriate coefficients. This argument can be extended byinduction to the case of N > 3 fields.An immediate and important corollary of Eq. (C10) is that

there are no corrections to S which are linear in αi j with i , j.Moreover, by looking at the second and third lines of Eq. (C10)we conclude that couplings give quadratic corrections only ifαi j , 0 and αji , 0 for i , j. In conclusion, a perturbativecoupling between nondegenerate operators Li and Lj gives atmost quadratic corrections to the QNM frequencies.

Appendix D: Degenerate spectra

There is a well-known analogy between QNMs for the po-tentialV and quasibound states for the potential−V [86]. Whenwe consider perturbations of the effective potential, followingthis analogy andRefs. [87, 88], we can computeQNMfrequen-cies by applying quantum mechanical perturbation theory. It

is reasonable that these considerations should extend to thecoupled system considered in this paper. In this appendix weshow that we can use quantummechanical perturbation theoryfor a coupled system which is degenerate at zeroth order. Wemostly follow standard notation from nonrelativistic quantummechanics.Let H0 be a Hamiltonian with a degenerate eigenvalue ω0

and two corresponding eigenstates |ω0,+〉, |ω0,−〉. A genericeigenstate with eigenvalue ω0 is given by the linear superpo-sition

|ω0〉 = c1 |ω0 ,+〉 + c2 |ω0 ,−〉 . (D1)

We denote the states by “±” indices rather than numerical in-dices because we are mainly interested in polar and axial grav-itational perturbations, but our discussion below is generic.We wish to find the spectra of the operator H0 + δV , where

δV is a small correction to the Hamiltonian. Consider theeigenvalue problem

(H0 + δV)|ω〉 = ω |ω〉 . (D2)

To first order in perturbation theory, the eigenvalues and eigen-functions can bewritten asω = ω0+ω1 and |ω〉 = |ω0〉+ |ω1〉 ,where ω1 and |ω1〉 are first-order corrections. Then Eq. (D2)becomes a relation between first-order quantities:

(δV − ω1)|ω0〉 = (ω0 − H0)|ω1〉 . (D3)

Using the relation 〈ω0,±|(ω0 − H0) = 0 and acting to the leftwith 〈ω0,±|, one obtains

c+δV++ + c−δV+− − ω1c+ = 0 (D4)c+δV−+ + c−δV−− − ω1c− = 0, (D5)

where δV±± := 〈ω0,±|δV |ω0,±〉, and similar relations definethe off-diagonal terms. This can be written in matrix form as(

δV++ − ω1 δV+−δV−+ δV−− − ω1

) (c+c−

)= 0. (D6)

To have a nontrivial solution for c±, the determinant must bezero. This yields Eq. (28).Fortunately, while theQNM frequency correctionω1 is non-

linear in the expectation values, the expectation values them-selves are linear. If we expand the potential as in Eq. (3), thenthe expectation values are given by

δV±± =∞∑k=0

α(k)±±〈ω0,±| f (r)

rk−2H

rk|ω0,±〉

=

∞∑k=0

α(k)±±δV

(k)±± , (D7)

δV±∓ =∞∑k=0

α(k)±∓〈ω0,±| f (r)

rk−2H

rk|ω0,∓〉

=

∞∑k=0

α(k)±∓δV

(k)±∓ . (D8)

13

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