Wiggly tails: A gravitational wave signature of massive fieldsaround black holes

Juan Carlos Degollado1,2 and Carlos A. R. Herdeiro11Departamento de Física da Universidade de Aveiro and I3N, Campus de Santiago,

3810-183 Aveiro, Portugal2Departamento de Ciencias Computacionales, Centro Universitario de Ciencias Exactas e Ingeniería,

Universidad de Guadalajara, Avenida Revolución 1500,Colonia Olímpica C.P. 44430 Guadalajara, Jalisco, México(Received 14 August 2014; published 16 September 2014)

Massive fields can exist in long-lived configurations around black holes. We examine how thegravitational wave signal of a perturbed black hole is affected by such “dirtiness” within linear theory. As aconcrete example, we consider the gravitational radiation emitted by the infall of a massive scalar field intoa Schwarzschild black hole. Whereas part of the scalar field is absorbed/scattered by the black hole andtriggers gravitational wave emission, another part lingers in long-lived quasibound states. Solvingnumerically the Teukolsky master equation for gravitational perturbations coupled to the massiveKlein-Gordon equation, we find a characteristic gravitational wave signal, composed by a quasinormalringing followed by a late time tail. In contrast to “clean” black holes, however, the late time tail containssmall amplitude wiggles with the frequency of the dominating quasibound state. Additionally, an observerdependent beating pattern may also be seen. These features were already observed in fully nonlinearstudies; our analysis shows they are present at linear level, and, since it reduces to a 1þ 1 dimensionalnumerical problem, allows for cleaner numerical data. Moreover, we discuss the power law of the tail andthat it only becomes universal sufficiently far away from the dirty black hole. The wiggly tails, by constrast,are a generic feature that may be used as a smoking gun for the presence of massive fields around blackholes, either as a linear cloud or as fully nonlinear hair.

DOI: 10.1103/PhysRevD.90.065019 PACS numbers: 11.15.Bt, 04.30.-w, 95.30.Sf

I. INTRODUCTION

The forthcoming science runs of the second generationgravitational wave (GW) detectors [1], and, in parallel, theuse of pulsar timing arrays [2], are promising a first directdetection of GWs from astrophysical sources before thedecade is over. Such detection will initiate the field of GWastrophysics, a natural realm for testing general relativity inthe strong field dynamical regime, as well as for con-straining theoretical models that predict new gravitationalinteractions [3]. This new field should play a significantrole over the next decades, delivering ever increasingprecision measurements. Understanding such measure-ments requires theoretical guidance. As such, unveilingtheoretical GW signatures of physical phenomena, andespecially of new physics, is particularly timely.Black holes (BHs) are the most compact objects pre-

dicted by general relativity. When involved in dynamicalprocesses, they play a role as GW sources. The GWemission pattern from a “clean” (i.e. vacuum) perturbedBH has long been identified. The BH relaxes back toequilibrium via damped oscillations in characteristicfrequencies—quasinormal ringing [4]—determined onlyby the final equilibrium state. A GW detector far from theBH will not only measure this ringing but also a late timetail [5], due to multiple scatterings by the spacetimecurvature of the GWs coming from the perturbed system.

This late time tail has therefore the ability to probe thespacetime in the vicinity of the BH.How does this simple picture change if the BH is “dirty”?

That is, if it is surrounded by some matter/fields? In a recentthorough examination [6,7], it was argued that the quasi-normal ringing will be unchanged, but that resonancescould occur in the late time behavior of waveforms. In thispaper we shall present a clean resonance which is an effectin GW physics that may be used to identify a specific kindof dirtiness.We consider a simple and tractable model of a dirty

environment: a BH surrounded by amassive field. The field’smass allows the existence of gravitationally trapped, long-lived field configurations around theBH, dubbed quasiboundstates, characterized by precise complex frequencies.Actually, massive field configurations can even becomeinfinitely long-lived in the case of Kerr BHs, for a specificfrequency determined by the horizon’s angular velocity[8–12]. The dynamical interaction of massive fields withBHs has been mostly discussed in the literature for scalarfields, both in the test field approximation (e.g. [13–15]) andin the fully nonlinear regime [16,17]. Our study also takes ascalar field as an illustrative case, but similar statements willhold for other massive fields, e.g. Proca fields.We show that the GW late time tail that follows the

quasinormal ringing of the BH contains small amplitude

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oscillations with the (real part of the) frequency of thedominating quasibound state that endows the BH with adirty environment. These oscillations, which were firstobserved in [16] solving the nonlinear Einstein-Klein-Gordon system, are an imprint left by the field’s cloudon the scatterings that originate the tail. Our setup, based onlinear perturbation theory, confirms the generic behaviordescribed in [16], and clarifies it is essentially controlled bylinear perturbations. Furthermore, the use of 1þ 1 evolu-tions allows us to optimize computational resources toextract physical signals.The existence of these wiggly tails relies on two

ingredients: a perturbed BH that relaxes by emittingGWs and a dirty environment around the BH providedby a long-lived configuration of a massive field, with onedominating oscillation frequency. This suggests that thesetails may be considered as strong evidence for the existenceof massive fields around BHs.We also discuss the power law decay of both the GWand

scalar field tails. These have known universal exponents forclean BHs [18]. For the scalar case we find a generic decayas RðsÞ ∼ t−2.5 as long as there are no quasibound states withsignificant amplitude at the extraction point. This is inagreement with the results in Ref. [13]. Otherwise, thescalar field decay is not a universal power law any longer.For the GW tail, we find that, as long as the extraction pointis sufficiently far away from the dirty BH, it decays asRðgÞ ∼ t−1.5. This is a different behavior from clean BHs, forwhich the Price decay law is t−ð2lþ3Þ with l ¼ 2 [5].Otherwise, the GW decay is affected by the dirtiness and nogeneric behavior could be unveiled.This paper is organized as follows: after this introduction

we describe the scalar field in the background of a BH inSec. II. In Sec. III we introduce the linear gravitationalperturbation equations in terms of the Weyl scalar Ψ4 aswell as the harmonic decomposition to get a set of 1þ 1coupled partial differential equations. In Sec. IV wedescribe the numerical code used to solve them and thenumerical results. Finally, in Sec. V we give some con-cluding remarks.

II. A MODEL FOR A DIRTY PERTURBEDBLACK HOLE

As a simple model of a perturbed BH in a dirtyenvironment we shall consider the interaction of aSchwarzschild BH with a scalar field Φ, with mass μand stress-energy tensor

Tμν ¼ Φ;μΦ;ν −1

2gμνðΦ;σΦ;σ þ μ2Φ2Þ: ð1Þ

The conservation of the stress-energy tensor implies thatthe field obeys the Klein-Gordon equation □Φ ¼ μ2Φ. Wewrite the background geometry in ingoing Kerr-Schildcoordinates, which are horizon penetrating and hence more

adequate for a numerical treatment. The corresponding lineelement is

ds2 ¼ −fðrÞdt2 þ 4Mr

drdtþ�1þ 2M

r

�dr2 þ r2dΩ2;

ð2Þwhere fðrÞ ≡ ð1 − 2M

r Þ and dΩ2 is the standard lineelement on S2.In order to find the solutions of interest of the Klein-

Gordon equation, the scalar field Φ can be expanded inFourier modes of frequency ω and spherical harmonicsYl;m0 with spin weight zero. Requiring that the field is

ingoing at the horizon and approaches zero at spatialinfinity leads to a discrete set of complex frequencies,for each l, corresponding to a fundamental mode andovertones. These solutions are the quasibound states. Thesestates have well-known frequencies of oscillation and ratesof decay, cf. for instance [19–21], which depend on thevalue of μ. The trend to keep in mind, for the Schwarzschildbackground, is that increasing the mass of the scalar fieldimplies decreasing the lifetime of the quasibound states,which is measured by the (inverse of the) imaginary part ofthe complex frequencies. For very small μ, e.g 10−23 eV, avalue compatible with some scalar field dark matter models[22], quasibound states may last for cosmological timescales around a supermassive BH, say with ∼108M⊙,without being significantly absorbed [14]. In such cases,the field configuration can be considered as a continuoussource of perturbation. Below we shall give specificfrequencies for the cases of interest herein, obtained usingthe continued fraction method first described in [23].In this paper, we study the gravitational perturbations of

the background sourced by the scalar field. These pertur-bations are evolved simultaneously with the Klein-Gordonequation in the unperturbed background, since the back-ground perturbations lead to second order effects in theKlein-Gordon equation. The model is the following. Weperturb the geometry by setting, at some initial time, a wavepacket of the scalar field in the background (2). A genericwave packet contains a range of frequencies that includesquasibound state frequencies but also contains enoughdynamics to awake the quasinormal modes of the BH.Thus, in the evolution of the scalar field wave packet, thereis a part which lingers around the BH in one dominatinglong-lived quasibound state. Other subleading quasiboundstates are also present and lead to one observable effect, aswe shall describe. Here, long-lived means that the timescale for the decay of the quasibound state is much longerthan the time scale for the quasinormal ringing.

III. SCALAR FIELD SOURCEDGRAVITATIONAL PERTURBATIONS

To compute the GWs induced by the backreaction of thescalar field on the metric we use linear perturbation theory

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in the Newman-Penrose formalism. The radiative informa-tion is extracted from the Newman-Penrose scalarΨ4 ¼ −Cαβγδkαm�βkγm�δ, where Cαβγδ is the first orderperturbed Weyl tensor, the vectors ðlα; mα; kαÞ are elementsof a Newman-Penrose null basis [24,25] and m�α is thecomplex conjugate of mα.For distant observers, Ψ4 describes outgoing GWs. A

central observation, due to Teukolsky [26,27], is that for(background) spacetimes of Petrov type D the first orderperturbation of Ψ4 decouples from the perturbations of theother Newman-Penrose scalars. Furthermore, he showedthat the resulting wave equation can be separated in theKerr (and thus Schwarzschild) background in its angularand radial parts, by expanding the perturbation in a basis ofspheroidal harmonics with the appropriate spin weight.To solve numerically the radial perturbation equation

coupled to the Klein-Gordon equation we use the lineelement (2) and the null vector basis kμ ¼ ð1;−1; 0; 0Þ,

lμ ¼ 1

2ð2− fðrÞ; fðrÞ;0;0Þ; mμ ¼ 1ffiffiffiffiffi

2rp ð0;0;1; i cscθÞ:

The details of the derivation of the equation for theperturbed Weyl scalar Ψ4 are given in [28]. After decom-posing Ψ4 in terms of spin weighted spherical harmonics(with weight −2):

Ψ4 ¼1

r

Xl;m

RðgÞl;mðt; rÞY−2

l;mðθ;φÞ; ð3Þ

the radial-temporal perturbation equation reads

−�1þ 2M

r

�∂ttR

ðgÞl;m þ

�1 −

2Mr

�∂rrR

ðgÞl;m þ 4M

r∂rtR

ðgÞl;m

þ 2

�2

rþM

r2

�∂tR

ðgÞl;m þ 2

�2

r−Mr2

�∂rR

ðgÞl;m

þ�2Mr3

−ðl − 1Þðlþ 2Þ

r2

�RðgÞl;m ¼ 16πrSðRðsÞ

l;mÞ; ð4Þ

where the sources SðRðsÞl;mÞ are projections of the stress-

energy tensor (1) along the null tetrad; see for exampleEqs. (3.10)–(3.21) of Ref. [28].To solve the second order equation (4) we decompose it

into a first order system using the first order variables

ψ ðgÞl;m ≡∂rR

ðgÞl;m; ΠðgÞ

l;m ≡ rþ2Mr

∂tRðgÞl;m−2

Mrψ ðgÞl;m: ð5Þ

Then, the following first order equations are obtained:

∂tRðgÞl;m ¼ 1

rþ 2MðrΠðgÞ

l;m þ 2Mψ ðgÞl;mÞ; ð6Þ

∂tψðgÞl;m ¼ ∂r

�1

rþ 2MðrΠðgÞ

l;m þ 2Mψ ðgÞl;mÞ

�; ð7Þ

∂tΠðgÞl;m ¼ 1

rþ 2Mð2M∂rΠ

ðgÞl;m þ r∂rψ

ðgÞl;mÞ

þ 2

rðrþ 2MÞ2 ½ð2r2 þ 5Mrþ 4M2ÞΠðgÞ

l;m

þ ðrþ 4MÞð2rþ 3MÞψ ðgÞl;m�

þ�2Mr3

−ðl − 1Þðlþ 2Þ

r2

�RðgÞl;m − 16πrSðRðsÞ

l;mÞ:

ð8Þ

To compute the evolution of the source in (8), we mustfollow the dynamics of the scalar field, given by the Klein-Gordon equation. We write this equation in the coordinates(2) and decompose the field as

Φ ¼ RðsÞl;mðt; rÞY0

l;mðθ;φÞ: ð9Þ

Then a second order partial differential equation forRðsÞl;mðt; rÞ is obtained. In order to solve it, we perform a

first order decomposition, in close analogy with thatperformed for the gravitational perturbations, by definingψ ðsÞl;m ≡ ∂rR

ðsÞl;m and

ΠðsÞl;m ≡ rþ 2M

r∂tR

ðsÞl;m − 2

Mrψ ðsÞl;m: ð10Þ

With these first order variables the Klein-Gordon equationbecomes an evolution equation for ΠðsÞ

l;m,

∂tΠðsÞl;m ¼ 1

rþ 2Mð2M∂rΠ

ðsÞl;m þ r∂rψ

ðsÞl;mÞ

þ 2

rðrþ 2MÞ2 ðψðsÞl;m − ΠðsÞ

l;mÞ

þ�μ2 −

lðlþ 1Þr2

þ 2Mr3

�RðsÞl;m: ð11Þ

With the solution of the scalar field at each time step wereconstruct the stress-energy tensor of the scalar field (1)and then the source term in (8).

IV. NUMERICAL RESULTS

We numerically solved the coupled system (6)–(8) and(10) and (11), for the evolution of the gravitationalperturbations and of the scalar field. We have used themethod of lines with a third order Runge-Kutta timeintegration and finite differencing with sixth order stencils.Such accuracy is needed in order to capture correctly thelate time tail behavior. The use of horizon penetratingcoordinates allow us to set the inner boundary inside thehorizon. The outer boundary is typically located at r ¼1000M (although in some cases we set it at r ¼ 2000M inorder to avoid any contamination coming in from the

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boundary). At the last point of the numerical grid we set upthe incoming modes to zero.We chose as initial data a static Gaussian perturbation of

the scalar field, with the form RðsÞ1;0ð0; rÞ ¼ e−ðr−rgÞ2=σ2 . For

a typical run, the pulse was centred at rg ¼ 10M andσ ¼ 0.5M. The gravitational perturbation is initially set tozero, since we are interested in the gravitational waveformproduced as a response to the presence of the scalar field.This choice of initial data does not bias the result. It hasbeen observed, by numerically evolving arbitrary initialdata, that the appearance of quasibound states is generic[29]. This guarantees that our choice of initial data willcreate the dirty environment we seek and simultaneouslyexcite the quasinormal ringing of the BH; but other choiceswould also achieve the same goal. Indeed, we observe thatpart of the scalar field falls into the BH, part is scattered andanother part remains as quasibound states.Figure 1 exhibits the gravitational signal forMμ ¼ 0.48,

as measured by a sufficiently distant observer. As expected,the quasinormal ringing is followed by a late time tail. Wehave found that the dirtiness is negligible for the ringing, inagreement with the results in [6,7]: performing a sinusoidalfit of the first stages of the signal, we obtained that thequasinormal ringing has the frequency of the quadrupolar(l ¼ 2) gravitational mode ωQNM ¼ 0.373 − i0.0889 of aclean Schwarzschild BH [4,18,30].A careful inspection of the late time behavior, on

the other hand, unveils a novel feature: on top of the decayingbehavior there are oscillations with a very small amplitude—Fig. 1 (inset). The frequency of the oscillations in this wigglytail precisely correspond to the real part of the frequencyof thedominating, i.e. largest amplitude, quasibound state. In thiscase the dominating state is the first overtone, with frequencyω1 ¼ 0.4717 − i1.4501 × 10−3 [14].To clearly demonstrate the observation in the last para-

graph and simultaneously exhibit yet another effect that

may be present in the late time signal, we present a slightlydifferent value of the field’s mass Mμ ¼ 0.4, for which wemonitor the behavior of both the scalar field and the GWsignal—Fig. 2. In this figure, the GW signal has beenrescaled for better visualization. The first observation is thatthe late time GW signal (black solid line) presents, besidesthe aforementioned wiggles, a beating pattern. The figureshows that the beating in the GW signal is resonating thebeating observed in the scalar field. Beating patterns aretypical in systems with two dominating frequencies withcomparable amplitudes. In this case, the beating frequencycorresponds to the difference between the first and secondovertones. The second observation is that the frequency ofthe wiggles seen in the late time GW signal coincides withthe frequency of the dominating quasibound state(s). Thisis shown in the figure’s inset.Both of the reported effects seen in the late time tails—

the wiggles and the beating—depend on the value of thefield’s mass μ. For the wiggles, the mass yielding the largestamplitude is Mμ≃ 0.48. One may think of this optimalmass as a balance between two effects. Smaller μ increasesthe lifetime of the dirtiness; however, it also increases itsspatial dispersion, and thus decreases the local effect of theperturbation. For the beating, the dominating quasiboundstates should have comparable amplitudes and the differ-ence in frequencies should yield a period compatible withthe observation times. An empirical conclusion, for our setsof initial data, is that the beating is most visible for Mμ≃0.4 (Fig. 3).The existence of beating patterns for scalar fields around

BHs has been discussed previously in different models[15,16,31]. But the study herein first reports a GW signalclearly resonating with such beating. We notice, however,that the beating—unlike the wiggles—depends on theextraction radius. For instance, if the observation point isclose to any of the nodes of one of the states contributing to

200 500 1000t/M

1×10-9

1×10-8

1×10-7

1×10-6

1×10-5

R(g

)

700 750 800 850 900 950 1000

3×10-9

4×10-9

5×10-9

FIG. 1. GW signal (l ¼ 2) for Mμ ¼ 0.48 (observer at 500M).Small amplitude wiggles can be seen in the late time behavior, inan otherwise apparently power law tail.

1000 1200 1400

3×10-4

4×10-4

5×10-4

500 1000 1500 2000 2500 3000t/M

-2×10-3

-1×10-3

-5×10-4

0

5×10-4

R(s)

R(g)

FIG. 2 (color online). Scalar (l ¼ 1) and GW (l ¼ 2) signal forMμ ¼ 0.40 (observer at 70M). The dominant frequency of thescalar field is that of the first quasibound state overtone, withω1 ¼ 0.3955 − i2.2668 × 10−4; the next leading quasiboundstate is the second overtone, with ω2 ¼ 0.3975 − i1.0035 × 10−4.

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the beating, this effect vanishes both in the scalar field andin the GW counterpart. Furthermore, the beating is sensi-tive to the initial data, since, although quasibound states areexcited for generic initial data, the amplitudes of thesestates depend on the details of such data.Finally, we have observed that the late time behavior of

the GW signal has a power law when extracted sufficientlyfar away from the dirty BH—Fig. 4. If the extraction takesplace within the dirtiness, this is not the case, as illustratedby the curve withMμ ¼ 0.1 in Fig. 4, since for this value ofthe mass, the dominant quasibound state has a significantamplitude at the extraction position.

V. CONCLUDING REMARKS

We have shown that the presence of a massive field inlong-lived configurations around a BH leaves a distinctivesignature in the late time behavior of the GW signal whenthe BH is perturbed: a wiggly tail. The frequency of thesewiggles is determined by the mass of the BH. For stellarmass BHs, M ∼ 1M⊙ − 10M⊙, this frequency lies in therange ω ∼ 12.7 kHz–1.27 Hz, which, as for the corre-sponding quasinormal modes, falls in the bandwidth ofground base detectors such as aLIGO [32]. For super-massive BHs, on the other hand, M ∼ 106M⊙ − 109M⊙,ω ∼ 12.7 mHz − 12.7 μHz, entering the bandwidth of thespace based detectors such as eLISA [3]. The amplitude ofthe wiggles, however, is 104 times smaller than that of thecorresponding quasinormal modes. As such the detectionof such a signal will be a considerable technical challenge.Concerning the mass of the scalar field, the values

considered here, Mμ ∼ 0.4–0.5, correspond to μ ∼10−17–10−20 eV for supermassive BHs and μ ∼10−11–10−12 eV for stellar mass BHs. Whereas these valuesare extremely small when compared to the masses ofstandard model particles, such light particles have beensuggested in the context of high energy physics scenariosbeyond the standard model. For instance, they arisenaturally in string compactifications, i.e. the axiverse [33].Both the existence of wiggles and of a beating in the GW

tails are qualitative features that may be used to distinguisha dirty environment from an isolated BH. Interestingly, themere presence of a beating is significant, since it providesevidence for two comparable amplitude frequencies in thedirty environment. For instance, for a Kerr BH with scalarhair [9], there should be a clearly dominating frequency—that of the background scalar field—and hence no notice-able beating should occur.The observation of any of these features provides a

quantitative measure of the frequencies of the quasiboundstates involved in the dirtiness and hence of the mass of thefield surrounding the BH.

ACKNOWLEDGMENTS

We would like to thank Vitor Cardoso for useful com-ments on a draft of this paper. This work was partiallysupported by Grant No. NRHEP 295189 FP7-PEOPLE-2011-IRSES. J. C. D. acknowledges support from FCT viaProject No. PTDC/FIS/116625/2010.

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500 1000 1500 2000 2500 3000t/M

9×10-10

2×10-9

4×10-9

7×10-9

1×10-8

R(g

)Mµ = 0.40Mµ = 0.42Mµ = 0.44Mµ = 0.46Mµ = 0.48

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R(g

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Mµ = 0.10Mµ = 0.30Mµ = 0.48Mµ = 0.50ta

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