Constraining modified gravity with ringdown signals: an explicit example

Jiahui Bao,1 Changfu Shi,1, 2 Haitian Wang,1 Jian-dong Zhang,1, ∗ Yiming Hu,1 Jianwei Mei,1, 2, † and Jun Luo1, 2

1TianQin Research Center for Gravitational Physics and School of Physics and Astronomy,Sun Yat-sen University (Zhuhai Campus), 2 Daxue Rd., Zhuhai 519082, P. R. China.

2MOE Key Laboratory of Fundamental Physical Quantities Measurement & Hubei Key Laboratory of Gravitation and QuantumPhysics, PGMF and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, P. R. China.

(Dated: October 18, 2019)

An explicit example is found showing how a modified theory of gravity can be constrained withthe ringdown signals from merger of binary black holes. This has been made possible by the fact thatthe modified gravitational theory considered in this work has an exact rotating black hole solutionand that the corresponding quasi-normal modes can be calculated. With these, we obtain thepossible constraint that can be placed on the parameter describing the deviation of this particularalternative theory from general relativity by using the detection of the ringdown signals from binaryblack holes’s merger with future space-based gravitational wave detectors.

I. INTRODUCTION

General relativity (GR) has enjoyed much success inpassing all experimental tests to date [1]. However, theo-retical problems related to the black hole singularity andblack hole information and observation evidence on darkmatter and dark energy all indicate that GR may not bethe final theory of gravity. Numerous modified theories ofgravity have been proposed to study possible extensionsto GR [2, 3].

The first detection of gravitational wave (GW) byLIGO [4] has made new tests of GR possible. Quasi-normal modes (QNMs) [5, 6], which encode all informa-tion in the ringdown stage of a compact binary merger,is especially important for the purpose [7]. According tothe no-hair theorem which states that all black holes innature are Kerr black holes, the QNMs constituting theringdown signal from a binary black hole merger are com-pletely determined by the mass and spin of the remnantblack hole. With the detection of at least two QNMs,it is possible to test the no-hair theorem by checking ifthe frequencies and the damping times are the same asthose predicted by GR [8, 9]. A failure of the no-hairtheorem may indicate that either GR is not the correctgravitational theory or GR is correct but the remnant isnot described by the Kerr metric.

In this work, we are interested in testing a specific mod-ified theory of gravity with the ringdown signal emittedby a binary black hole’s merger. For this purpose, we willblame all possible failure of the no-hair theorem on thedifference between the modified gravity theory and GR.

There have been work on constraining modified theo-ries of gravity or no-hair theorem with GW observations.However, so far the constraints are mostly given in termsof phenomenological parameters whose relation to eachmodified theory of gravity is not known explicitly [10–12].In order to constrain a specific modified theory of gravity

∗ Emial:zhangjd9@mail.sysu.edu.cn† Emial:meijw@sysu.edu.cn

with ringdown signal, one has to overcome at least twoobstacles:

• The first is to have reliable information on at leasttwo QNMs through GW detection. GW150914 isthe first and by far the strongest GW signal de-tected, with a total signal to noise ratio (SNR)reaching 24. But the SNR for the ringdown stage isonly about 7, so it is extremely difficult to extractthe frequencies and damping times for the subdom-inant mode. For future space-based GW detectors,such as LISA [13] and TianQin [14], and the nextgeneration ground-based detectors, such as Ein-stein Telescope [15] and Cosmic Explorer [16], highenough SNR is possible.

• The second is to calculate the QNMs for rotatingblack holes in the modified theory of gravity understudy. The most promising source for testing theno-hair theorem with QNMs is the merger of mas-sive black holes, of which the remnant black holeis usually a rotating one. But rotating black holesolutions in modified theories of gravity are diffi-cult to find and it is also difficult to calculate thecorresponding QNMs. For example, QNMs haveonly been studied for non-rotating or slowly rotat-ing black holes in very few alternative theories, suchas the dynamical Chern-Simons gravity [17, 18],Einstein-dilaton-Gauss-Bonnet gravity [19, 20] andHorndeski gravity [21].

Due to these difficulties, an example has been lack-ing where the ringdown signals from binary black holes’merger are used to constrain a specific modified theoryof gravity.

We find that in the Scalar-Tensor-Vector Gravity the-ory (STVG) [22], a rotating black hole solution is knownand the dependence of the corresponding QNMs on thekey STVG parameter can also be obtained by simplemeans. As such, one can place explicit constraint onSTVG with the ringdown signals from binary black holes’merger to be detected with future GW detectors, forwhich we will focus on TianQin [14] and LISA [13].

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The paper is organized as follows. In section II, weintroduce some basics of STVG, including the rotatingblack hole solution already known, then we obtain thecorresponding QNMs. In section III, we study how theGR deviating parameter in STVG can be constrainedwith TianQin and LISA using the ringdown signal fromthe merger of binary black holes. In section IV, we givea brief discussion and summary.

II. ROTATING BLACK HOLE SOLUTION INSTVG AND ITS QNM

The action of STVG is given by [22]

S =

∫d4x(LG + Lφ + LS + LM ), (1)

where LM is the Lagrangian density of matter, and

LS =√−g

1

G3

[1

2gµν∇µG∇νG− V (G)

]+

1

µ2G

[1

2gµν∇µµ∇νµ− V (µ)

],

LG =1

16πG

√−g(R+ 2Λ) ,

Lφ =1

4π

√−g[1

4BµνBµν −

1

2µ2φµφµ + Vφ(φ)

], (2)

are the Lagrangian densities of the scalar, tensor and vec-tor fields, respectively. The fields G(x), µ(x) are scalarsrelated to Newton’s constant GN and the mass of thevector field φµ, respectively, Bµν = ∂µφν − ∂νφµ, andV (G), V (µ), Vφ(φ) are potentials.

A rotating black hole solution in the theory has beenconstructed for the special case G = GN (1 + α), µ ≈ 0and V (G) = V (µ) = Vφ(φ) = 0 . In this case the actionreduces to that of the Einstein-Maxwell theory,

S =1

16πG

∫d4x√−g(R+ 2Λ +GBµνBµν

), (3)

and the rotating black hole solution is nothing but theKerr-Newman solution with a special choice of its chargeparameter [23]:

ds2 = −∆S

ρ2(dt− a sin2θdφ)2 +

ρ2

∆Sdr2

+sin2θ

ρ2[(r2 + a2)dφ− adt]2 + ρ2dθ2 ,

∆S = r2 − 2GMr + a2 +GQ2φ ,

ρ2 = r2 + a2cos2θ , (4)

where a = J/M with J being the angular momentum.The conserved charge of the vector field is assumed to beproportional to the mass [23]

Qφ = ±(αGN )1/2M . (5)

The action (3) differs from that of GR in two ways.Firstly, the vector field, φµ, in (3) is to be distinguishedfrom the usual electromagnetic field. Secondly, the cou-pling constant G is different from Newton’s gravitationalconstant GN . In the case of α = 0, the vector field φµvanishes, G returns to GN and the gravitational pertur-bation of (4) returns to that of a Kerr black hole in GR.By studying the QNMs of the ringdown signal, one canimpose constraint on the GR deviating parameter α .

The QNMs of the Kerr-Newman black hole have beenstudied with various methods, including a considera-tion of the static case [24]. Dias et al. have stud-ied the QNMs of the Kerr-Newman black hole with theNewton-Raphson method, where they solved the per-turbation equations directly without variable separation[25]. Dudley and Finley (DF) have obtained approxi-mate decoupled equations for gravitational perturbationsof the Kerr-Newman black hole [26]. With these equa-tions, Berti and Kokkotas have computed the QNMswith both the WKB and the continued fraction method[27, 28]. The QNMs of the weakly charged [29] and theslow-rotation [30, 31] solutions have also been calculated.The Newton-Raphson method is relatively more precise,but solving the DF equations with the continued fractionmethod is more efficient and we shall adopt this approachin the present work.

To test STVG, we need to find out the dependenceof QNMs’ complex frequencies Ωlm on the parameter α.Following [27], the DF equations of (4) are given by

0 =d

du

[(1− u2)

dSlmdu

]+

[a2Ω2

lmu2

−2aΩlmsu+ s+ Elm −m+ su

1− u2

]Slm ,

0 = ∆−sSd

dr

[∆s+1S

dR

dr

]+

1

∆S

[K2 − isd∆S

drK

+∆S

(2is

dK

dr− Elm − a2Ω2

lm + 2amΩlm

)]R ,(6)

where u = cosθ, K = (r2 + a2)Ωlm − am, and Elm isthe separation constant. Note s = −2 corresponds to thegravitational perturbation. Imposing the purely ingoingand outgoing boundary conditions at the black hole hori-zon and the space infinity, respectively, the angular func-tion Slm and the radial function R can be constructedas

Slm(u) = eaΩlmu(1 + u)|m−s|/2(1− u)|m+s|/2

×∞∑n=0

an(1 + u)n ,

R(r) = eiΩlmr(r − r−)−1−s+iΩlm+iσ+(r − r+)−s−iσ+

×∞∑n=0

dn

(r − r+

r − r−

)n, (7)

where r± = GM ± (G2M2 − a2 − GQ2φ)1/2 are the two

roots of ∆S = 0, and σ+ = [Ωlm(2GMr+ − GQ2φ) −

am]/(r+ − r−) .

3

Plugging (7) into (6), one can obtain two three-termcontinued fraction relations, which can be solved numer-ically for Ωlm and Elm for each QNM. This methodworks more reliably for Qφ ≤ M/2 [27], correspondingto α ∈ [0, 1/4]. Frequencies for the fundamental modeswith l = m = 2 are listed in TABLE I, where we haveset GN = M = 1 . We can find that it is exactly thesame result of Kerr-Newman if we replace α with Qφ by(5). This is an obvious result since the metric and theequation are the same as Kerr-Newman with this replace-ment. Besides, if α = 0 we can recover the QNMs resultof Kerr black hole.

An illustration of the dependence of the deviationsof QNM frequencies for l = m = 2 mode on the cou-pling constant α and final spin χf is given in FIG. 1,where δωlm = [ωlm(α) − ωlm(0)]/ωlm(0) and δτlm =[τlm(α)− τlm(0)]/τlm(0) . These are the parameters usedin theory-agnostic parameterized framework which de-scribe the frequency deviations on GR results. Suchframework has been discussed in several work, such as[10–12, 32–34]. We can find that for special values ofspin, the deviations of ω22 and τ22 are almost propor-tional to α. On the other hand, for small χf the relativedeparture is almost the same for fixed α. But for largeχf such as χf > 0.6, the deviation for τ22 will grow withχf , and the deviation for ω22 will decrease with χf . Thedeviations for other modes are similar to 22 mode, thuswe do not show them here.

For later convenience, we decompose the complexquasi-normal frequency into real and image part asΩlm = ωlm + i/τlm, and fit the numerical result witha set of phenomenological formulae similar to those in[8], the

Mωlm = f1(1− χf )f2 + f3 + α[f4(1− χf )f5 + f6] ,

Qlm = q1(1− χf )q2 + q3 + α[q4(1− χf )q5 + q6] ,(8)

where Qlm = ωlmτlm/2, and χf = J/(GM2) is the di-mensionless spin of the remnant black hole. The con-stants fi and qi are listed in TABLE II, where the lastcolumn indicates the maximal percentage of error of thefit formulae from the true values.

III. CONSTRAINING STVG WITH TIANQINAND LISA

In this section, we study how the GR deviating param-eter α in STVG can be constrained using the ringdownsignal from the merger of massive black holes to be de-tected with future space-based GW detectors, focusingon TianQin [14] and LISA [13].

A. Detectors

The first detector we consider is TianQin [14], whichbe a constellation of three satellites on a geocentric orbitwith radius about 105 kilometers. We adopt the following

χf = 0.8

χf = 0.8δω22 (χf = 0.2, 0.6, 0.7, 0.8)

δτ22 (χf = 0.2, 0.6, 0.7, 0.8)

0.00 0.05 0.10 0.15 0.20 0.25

-0.1

0.0

0.1

0.2

0.3

0.4

α

Relativedeparture

α = 0.25

α = 0.25

δω22 (α = 0.1, 0.15, 0.2, 0.25)

δτ22 (α = 0.1, 0.15, 0.2, 0.25)

0.0 0.2 0.4 0.6 0.8

-0.1

0.0

0.1

0.2

0.3

0.4

χfRelativedeparture

FIG. 1. Dependence of the relative departures δw22 and δτ22on α(top) and χf (bottom). The lighter lines correspond tosmaller values for the fixed parameter.

model for the sky averaged sensitivity of TianQin [10, 14,35, 36],

Sn(f) =10

3

[ 4Sa(2πf)4L2

0

(1 +

10−4Hz

f

)+Sx(f)

L20

]×[1 +

(2fL0/c

0.41

)2], (9)

where L0 =√

3 × 108m, c is the speed of light,√Sa =

1×10−15ms−2Hz−1/2 is the average residual accelerationon each test mass and

√Sx = 1× 10−12 mHz−1/2 is the

total noise of displacement measurement in a single laserlink. To avoid the problem of having telescopes pointingto the Sun, TianQin adopts a “3 month on + 3 month off”observation scheme. To fill up the observation gap, onemay consider having a twin set of TianQin constellationsto operate consecutively. Such a scheme will not affectthe sensitivity of each detector.

The second detector we consider is LISA [13], the con-cept of which has been around for several decades. ForLISA, we shall use the sensitivity curve given in [37].

4

TABLE I. Frequencies for the fundamental quasi-normal modes with l = m = 2.

a α=0 α=0.05 α=0.1 α=0.15 α=0.2 α=0.25

0 0.373672 -0.088962 0.359461 -0.085110 0.346353 -0.081574 0.334221 -0.078316 0.322955 -0.075305 0.312465 -0.072513

0.2 0.402145 -0.088311 0.387457 -0.084463 0.373892 -0.080928 0.361322 -0.077670 0.349636 -0.074657 0.338741 -0.071863

0.4 0.439842 -0.086882 0.424807 -0.083012 0.410909 -0.079453 0.398019 -0.076169 0.386025 -0.073128 0.374834 -0.070303

0.6 0.494045 -0.083765 0.479241 -0.079765 0.465594 -0.076068 0.452972 -0.072637 0.441267 -0.069443 0.430384 -0.066459

0.8 0.586017 -0.075630 0.574636 -0.070848 0.564788 -0.066266 0.556428 -0.061827 0.549555 -0.057467 0.544233 -0.053118

0.96 0.767674 -0.049434 - - - - -

TABLE II. Constants and maximal error for (8).

lm f1 f2 f3 f4 f5 f6 %

22 -1.2749 0.1197 1.6388 0.0615 5.2626 -0.2592 3.13

21 -0.3194 0.2712 0.6856 0.0448 4.8229 -0.2554 1.99

33 -1.3078 0.1894 1.8907 0.1244 4.1520 -0.4466 3.56

44 -2.7346 0.1109 3.5309 0.0698 6.4496 -0.5616 2.89

lm q1 q2 q3 q4 q5 q6 %

22 1.4422 -0.4941 0.6693 0.3268 -1.6119 -0.1761 1.64

21 1.6886 -0.2919 0.3941 -1340.6 0.0007 1340.7 8.97

33 2.3010 -0.4837 0.9527 0.5876 -1.5442 -0.4343 2.96

44 3.0104 -0.4910 1.3079 0.6208 -1.6737 -0.2787 1.54

B. Waveform of the ringdown signal

The ringdown waveform is given by

h+(t) =Mz

DL

∑l,m>0

Alme−t/τlm

×Y lm+ cos(ωlmt−mφ) ,

h×(t) = −Mz

DL

∑l,m>0

Alme−t/τlm

×Y lm× sin(ωlmt−mφ) , (10)

where Mz is the red-shifted mass of the remnant blackhole, DL is the luminosity distance to the source, Y lm+,× =

Y lm+,×(ι) is the sum of spin -2 weighted spherical har-monics [38], with ι ∈ [0, π) being the angle between thespin-axis of the source and the line-of-sight to the source,φ ∈ [0, 2π] is the initial orbital phase of the source, andAlm is the amplitude of the lm mode [12]

A22(ν) = 0.864ν ,

A21(ν) = 0.43[√

1− 4ν − χeff ]A22(ν) ,

A33(ν) = 0.44(1− 4ν)0.45A22(ν) ,

A44(ν) = [5.4(ν − 0.22)2 + 0.04]A22(ν) , (11)

where ν = m1m2/(m1 +m2)2 is symmetric mass ratio,

χeff =1

2

[√1− 4νχ1 +

m1χ1 −m2χ2

m1 +m2

]is the effective spin, and m1,m2, χ1, χ2 are the massesand dimensionless aligned spins of the binary black holebefore merger. We use four leading modes to constructthe ringdown signal, which are 22, 21, 33, 44 mode respec-tively. The chosen of modes follows [10–12, 33], and moredetailed discussions can be found therein.

C. Statistical method

The SNR for a GW signal is obtained with the follow-ing formula,

SNR[h] =√

(h|h) , (12)

and the inner product for any pair of signals p(f) andq(f) is defined as

(p|q) = 2

∫ fhigh

flow

p∗(f)q(f) + p(f)q∗(f)

Sn(f)df , (13)

where flow is taken to be half the frequency of the (2, 1)mode and fhigh is taken to be twice the frequency of the(4, 4) mode, to prevent the “junk” radiation that occursin the Fourier transformation [11].

In the case of large SNR, the uncertainty in parameterestimation is given by

∆θa ≡√〈δθaδθa〉 ≈

√(Γ−1)aa (14)

where θa are parameters to be estimated, 〈. . . 〉 denotesthe expectation value, and Γ−1 is the inverse of the Fisherinformation matrix (FIM),

Γab =( ∂h∂θa

∣∣∣ ∂h∂θb

). (15)

We will focus on the sky-averaged result and the pa-rameter space to be considered is

−→θ = M,χf , DL, ν, χeff , t0, φ, ι, α, (16)

where t0 is the time of coalescence.

5

It is also interesting to consider the combined con-straint from all events that can be detected throughoutthe lifetime of a detector. Assuming that all the detectedevents are independent of each other, we can construct acombined FIM to study the cumulative constraint on α.The parameter space is

−→θ = Mz1...Mzi ;χf1...χfi ;DL1...DLi ; ν1...νi ;

χeff1...χeffi ; t01...t0i ;φ1...φi ; ι1...ιi ;α . (17)

D. Results

The projected constraint of TianQin and LISA on theGR deviating parameter α with the detection of a singlemassive black hole merger is illustrated in FIG. 2 andFIG. 3, respectively. In both figures, δα indicates thedeviation of α from 0, and other parameters are: DL =15Gpc, ν = 2/9, χeff = 0.3, ι = π/3 and t0 = φ = 0.One can see that, α is best constrained with TianQin forMz ∼ 4× 106M and with LISA for Mz ∼ 107M .

0.001

0.01

0.1

0.1

1

5.0 5.5 6.0 6.5 7.0 7.5 8.00.0

0.2

0.4

0.6

0.8

Lg[Mz/M⊙]

χf

FIG. 2. The projected constraint of TianQin on δα, varyingwith χf and Mz.

In optimal scenarios, both LISA and TianQin are ex-pected to detect hundreds of massive black hole mergersthroughout their mission lifetime [36, 39–41]. The pro-jected number of massive black hole mergers that canbe detected are largely model dependent. We use threemodels for the merger history of massive black holes ashas been considered in [36]. These models are denotedas “popIII”, “Q3 d” and “Q3 nod”, corresponding to thelight seed model [42] and the heavy seed models [43–45]with and without time delay between the merger of mas-sive black holes and that of their host galaxies, respec-tively. Further explanation of these models can be foundin [36] and references therein.

We shall consider several different detector scenar-ios, including TianQin operating for a nominal lifetimeof 5 years (“TQ”), a twin set of TianQin constellation

0.001

0.01

0.1

0.1

1

10

5.0 5.5 6.0 6.5 7.0 7.5 8.00.0

0.2

0.4

0.6

0.8

Lg[Mz/M⊙]

χf

FIG. 3. The projected constraint of LISA on δα, varying withχf and Mz.

operating for 5 years (“TQ tc”), LISA operating for4 years (“LISA 4y”) and LISA operating for 10 years(“LISA 10y”).

For each of the detector scenarios, we produce 100 sim-ulated catalogue from each of the models for the mergerhistory of massive black holes. Each simulated catalogueis consisted of all the events that can be detected withthe corresponding detector scenario (selected if the SNRof the whole waveform is greater than 8). Each data setgives a combined constraint on the GR deviating param-eters α. For a given detector scenario, one can averageover the corresponding 100 sets of data to obtain an av-eraged constraint on α. The results are listed in TABLEIII.

TABLE III. The projected constraint on δα with differentdetector scenarios coupled with various models for the growthand merger history of massive black hole binaries. See textfor further explanation.

popIII Q3 nod Q3 d

TQ 0.0331 ± 0.0222 0.0063 ± 0.0027 0.0101 ± 0.0054

TQ tc 0.0181 ± 0.0114 0.0047 ± 0.0017 0.0066 ± 0.0031

LISA 4y 0.0066 ± 0.0029 0.0020 ± 0.0008 0.0041 ± 0.0016

LISA 10y 0.0037 ± 0.0021 0.0011 ± 0.0006 0.0026 ± 0.0001

IV. SUMMARY AND DISCUSSION

To sum up, we have presented an explicit exampleof using the ringdown signal from a binary black holemerger to constrain a modified theory of gravity, i.e.STVG. We find that both TianQin and LISA have thepotential to constrain α to the level of a few percent orbetter.

6

There is a caveat with the result obtained. Since boththe action (3) and the solution (4) are essentially thesame as those of a charged rotating black hole, the ef-fect of α in STVG is degenerate with that of the electriccharge of a Kerr-Newman black hole in a usual Einstein-Maxwell system. However, the electric charges of astro-physical black holes tend to be quickly reduced due to thequantum Schwinger pair-production effect [46, 47] andthe vacuum breakdown mechanism [48–50]. E. Barausseet al. [51] have presented a theoretical upper bound onthe charge-to-mass ratio of black holes, Q/M ∼ 10−3,corresponding to α ∼ 10−6 . So if future space-basedGW detectors were to consistently find α significantly

greater than the order of 10−6 , the result is more likelydue to a genuine STVG effect rather than the electriccharges of black holes.

ACKNOWLEDGMENTS

We thank Peng-Cheng Li, Yi-Fan Wang and ViktorT. Toth for helpful discussion and correspondence. Wealso thank Alberto Sesana and Enrico Barausse for shar-ing their simulated catalogue of massive black holes.This work has been supported by the Natural ScienceFoundation of China (Grant Nos. 91636111, 11690022,11703098, 11805286, 11475064).

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