Testing No-Hair Theorem by Quasi-Periodic Oscillations: the quadrupole of GRO J1655 − 40
PPrepared for submission to JCAP
Testing No-Hair Theorem byQuasi-Periodic Oscillations: thequadrupole of GRO J1655 − Alireza Allahyari, a Lijing Shao b,c a School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P. O. Box19395-5531, Tehran, Iran b Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China c National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, ChinaE-mail: [email protected], [email protected]
Abstract.
We perform an observational test of no-hair theorem using quasi-periodic oscil-lations within the relativistic precession model. Two well motivated metrics we apply areKerr-Q and Hartle-Thorne metrics in which the quadrupole is the parameter that possibly en-codes deviations from the Kerr black hole. The expressions for the quasi-periodic frequenciesare derived before comparing the models with the observation. We encounter a degeneracy inconstraining spin and quadrupole parameters that makes it difficult to measure their values.In particular, we here propose a novel test of no-hair theorem by adapting the Hartle-Thornemetric. It turns out that a Kerr black hole is a good description of the central object in GROJ1655 −
40 given the present observational precisions. a r X i v : . [ g r- q c ] F e b ontents δ -metric 32.1.1 Mass and quadrupole 42.1.2 Kerr- Q Metric 52.1.3 QPOs for Kerr- Q metric 52.2 Rotating Hartle-Thorne metric 82.2.1 QPOs for rotating Hartle-Thorne metric 9 Q metric 103.2 Hartle-Thorne metric 123.2.1 Model 1 123.2.2 Model 2 13 Black holes (BHs) are an indispensable prediction of the theory of general relativity (GR).Observationally they appear in various astrophysical environments and there is wealth ofdirect or indirect evidence pointing towards the existence of supermassive BHs, with massesas large as 10 –10 M (cid:12) . It is believed that the centers of most sufficiently massive galaxiesharbour supermassive BHs. In the current paradigms for quasars and active galactic nuclei, aswell as for the formation of galaxies, the presence of supermassive BHs is of crucial importancein the center of almost every galaxy [1].A generic prediction of GR is that BHs are understood to constitute the end state ofgravitational collapse of matters. In GR, a (stationary) BH final state is given in general bythe charged Kerr spacetime characterized by its mass M , electric charge e BH , and angularmomentum J . This is known as the no-hair theorem . According to this theorem the multiplemoments of a Kerr BH satisfy M l + i S l = M (i a ) l [2]. It is nonetheless desirable to findsolutions that admit more parameters. A test of the no-hair theorem can both identify theobserved dark compact objects with Kerr BHs and verify the validity of GR in the strong-fieldregime. Given the importance of such tests, many focused studies of the no-hair theoremhave been highly regarded by many authors [3–12].BHs are characterized by the existence of the event horizon, the boundary beyond whicheven light is unable to escape. The existence of the event horizon has the advantage that theexterior region satisfies the principle of causality. To preserve this, Penrose has conjecturedthat the singularities that would actually occur in nature must be BHs. This is known as theCosmic Censorship Conjecture. However, there is no general proof for this conjecture andmore general collapsed configurations are possible within the classical framework of GR.– 1 –he other evidence in favor of existence of BHs is the detection of gravitational wavesfrom astrophysical sources. Most detections of gravitational waves have been interpreted interms of binary BH mergers [13–18].Evidently, extremely compact gravitationally collapsed systems definitely exist, but theobservational evidence regarding their precise nature is somehow inconclusive. We thereforeturn to compare the theoretical evidence based on Einstein’s GR with observations.One yet promising arena to test the nature of the compact objects is by quasi-periodicoscillations (QPOs) observed in the X-ray flux emitted by accreting BHs in X-ray binarysystems [19, 20]. In these systems, a BH or a neutron star (NS) accretes materials froma stellar companion [21]. QPOs are observed as narrow features in the power spectra ofthe light curves of accreting NSs and BHs [22]. The very first hint towards their discoveryin the literature dates back to the results reported in Ref. [23] and a definite discovery inRef. [24]. There are various mechanisms proposed to account for such narrow features in thepower spectra ranging from relativistic precession models, diskoseismology models, resonancemodels and p -mode oscillations of an accretion torus [25–34]. These attempts have tried topin down the mechanism that could account for the QPO features.One of the highly regarded models for QPOs is the relativistic procession model ac-cording to which the QPO frequencies are believed to be related to fundamental frequenciesof test particles orbiting a central object [25, 27]. These particles undergo small oscillationswhen perturbed as they revolve around the central object. This model was later extendedin Ref. [35]. The promising aspect of using the QPOs as a method to study the nature ofcompact objects is that they can be measured with rather high precision [36–41]. The crucialfact about QPOs is that they can be used to probe the strong filed regime of GR [42–46].Although it appears that modeling an accreting object with a Kerr metric is a goodapproximation, the exact nature of the compact object deserves a thorough investigation. Weseek to model an accreting object associated with the QPOs as a rotating metric with a smallquadrupole. Our models are based on the Kerr-Q metric and Hartle-Thorne metric [47–49].They represent a slowly rotating object with the quadrupole q and the physical mass M and the rotation a . In a previous paper, Allahyari et al. have investigated the propertiesof Kerr-Q metric and derived the quasi normal frequencies of this metric [47, 50]. Here, wewish to constrain the quadrupole of the central object in GRO J1655 −
40 by its observedQPOs [51].This manuscript is structured as follows. In Sec. 2.1 we give a general review of rotating δ -metric and expand it to first order in the quadrupole parameter and define QPOs forthis metric. In Sec. 2.2, we review the Hartle-Thorne metric and its QPOs. In Sec. 3,finally we constrain the quadrupole in our models by comparing with the observed QPOs forGRO J1655 −
40. Across this manuscript, for the signature of the spacetime metric, we willimplement the ( − , + , + , +) convention. In this section we introduce two metrics that admit a quadruple, namely the Kerr-Q metricand the Hartle-Thorne metric. They both describe a configuration to first order in thequadrupole and to second order in the rotation. q -metric is derived from the rotating δ -metric by writing δ = 1 + q and m = M/ (1 + q ) and expanding to first order in q where M is the physical mass as we will explain. If we repeat the same steps for rotating δ -metricand additionally expand to second order in the rotation parameter a , we get Kerr-Q metric– 2 –47, 50]. These metrics were derive from the exact solutions by expanding to the desiredorder. There is another solution with a quadrupole which is not derived from an exactsolution. This metric is the Hartle-Thorne solution. δ -metric We start with the δ -metric and explain how to derive the Kerr-Q metric. The rotating δ -metric is a generalized Kerr metric that admits a quadrupole. This metric is obtained byrotating the δ -metric as the seed metric using solution generating techniques involving theHKX transformations in which symmetries of the field equations are used to find solutionsthat admit with rotation, multipole moments and magnetic dipole from given seed metrics [48,52, 53].Supposing that the central object is axisymmetric, we focus on the metrics with thissymmetry. The general stationary axisymmetric line element can be represented in spheroidalcoordinates ( t, x, y, φ ) as ds = − f ( dt − ωdφ ) (2.1)+ σ f (cid:20) e γ ( x − y ) (cid:18) dx x − dy − y (cid:19) + ( x − − y ) dφ (cid:21) , where σ is a positive constant and all the metric functions, F , ω and γ , depend on x and y .In spheroidal coordinates they are f = AB ,ω = − (cid:18) − a + σ CA (cid:19) , (2.2)e γ = 14 (cid:16) mσ (cid:17) A ( x − q (cid:20) x − x − y (cid:21) (1+ q ) , where we have defined A = a + a − + b + b − , B = a + b , (2.3) C = ( x + 1) q (cid:2) x (1 − y )( λ + η ) a + + y ( x − − λη ) b + (cid:3) , (2.4)along with a ± = ( x ± q [ x (1 − λη ) ± (1 + λη )] ,b ± = ( x ± q [ y ( λ + η ) ∓ ( λ − η )] , (2.5)and λ = α ( x − − q ( x + y ) q , (2.6) η = α ( x − − q ( x − y ) q , with αa = m − σ . – 3 –e recover the q -metric when a tends to zero [50, 54]. We have also checked that when q = 0, we recover the Kerr metric in Boyer–Lindquist coordinates by using the followingtransformation x = r − mσ , (2.7) y = cos θ,σ = (cid:112) m − a . Finally we transform the metric to spherical coordinates. The metric in spherical coordinatescan be written as ds = − f dt + 2 f ωdtdφ + e γ f BA dr + r e γ B f dθ (2.8)+ (cid:20) r A sin θf − f ω (cid:21) dφ , where we have A = 1 − mr + a r , (2.9) B = 1 − mr + a r + σ sin θr . (2.10)Note that r σ A = − (cid:18) r − mσ (cid:19) . (2.11) For small values of a and q we may expand the metric to first order in q and to second orderin a . We also neglect terms like q a and q a . We have g tt = − ˆ A (cid:16) q ln ˆ A (cid:17) − a m cos θr , (2.12) g tφ = − am sin θr , (2.13) g rr = 1ˆ A (cid:32) q ln ˆ A ˆ B (cid:33) − a − cos θ ˆ A r ˆ A , (2.14) g θθ = r + a cos θ + qr ln ˆ A ˆ B , (2.15) g φφ = (cid:20) r − qr ln ˆ A + a (cid:18) m sin θr (cid:19)(cid:21) sin θ, (2.16)where ˆ A := 1 − mr , (2.17)ˆ B := 1 − mr + m sin θr . (2.18)– 4 –his metric represents the superposition of Kerr metric and q -metric. When a = 0, we getthe δ -metric expanded to first order in q [50]. When q = 0, we get the Kerr metric to secondorder in a . The mass and quadrupole of this metric have been found after taking the weakfield limit and a coordinate transformation by which the metric takes the Newtonian form.Therefore, we have for the physical mass, quadrupole, and angular momentum, M = m (1 + q ) , Q = 23 M q + a M, J = M a. (2.19)Therefore, the physical mass M is different from m . Also note that the parameter q givesthe quadrupole of the metric [47]. Following the discussion in Ref. [50], for oblate (prolate)configurations we have q > q < q >
0. Hence, we will assume q > Q Metric
Note that in the rotating δ -metric in Eqs. (2.12)–(2.16), the mass parameter m is not physical.We wish to adapt it to a more desirable form expanded in terms of the physical mass M . Tocast the metric in terms of the physical mass, we must take the rotating δ -metric and replace m by M/ (1 + q ). After expanding the resulting metric to linear order in q and to secondorder in a , we obtain ds = − (cid:20) ˜ A + q (cid:18) Mr ˜ A + ln ˜ A (cid:19) ˜ A + 2 a M cos θr (cid:21) dt − aM sin θr dt dφ (2.20)+ (cid:34) A − q ˜ A (cid:32) Mr ˜ A + ln ˜ B ˜ A (cid:33) − a − cos θ ˜ A r ˜˜ A (cid:35) dr + (cid:32) − q ln ˜ B ˜ A + a r cos θ (cid:33) r dθ + (cid:20) − q ln ˜ A + a r (cid:18) M sin θr (cid:19)(cid:21) r sin θ dφ , where ˜ A := 1 − Mr , (2.21)˜ B := 1 − Mr + M sin θr . (2.22)This metric reduces to the Kerr metric (to second order in a ) when q = 0. Let us emphasizethat this metric has different components from the rotating δ -metric in Eqs. (2.12)–(2.16),as we have rewritten it in terms of the physical mass M rather than the non-physical massparameter m . We will implement this metric in our calculations. Q metric Let us consider a test particle in orbit around a stationary axisymmetric source. The generalline element in spherical coordinates is given by ds = g tt dt + g rr dr + g θθ dθ + 2 g tφ dtdφ + g φφ dφ , (2.23)where all functions depend on r and θ . There are two constants of motion imposed by thesymmetries of the metric, the specific energy at infinity, E , and the z -component of specific– 5 –ngular momentum L z . Therefore, we have˙ t = Eg φφ + L z g tφ g tφ − g tt g φφ , (2.24)˙ φ = − Eg tφ + L z g tt g tφ − g tt g φφ , (2.25)where dot represents the derivative with respect to the affine parameter. For a timelike path, g µν ˙ x µ ˙ x ν = −
1. After using Eqs. (2.24) and (2.25), we have g rr ˙ r + g θθ ˙ θ = V eff ( r, θ, E, L z ) , (2.26)where the effective potential V eff is V eff = E g φφ + 2 EL z g tφ + L z g tt g tφ − g tt g φφ − . (2.27)We are interested in the equatorial circular orbits with θ = π/ r = ¨ r = ˙ θ = 0. Theradial component of the geodesic equations givesΩ φ = ˙ φ ˙ t = − ∂ r g tφ ± (cid:113) ( ∂ r g tφ ) − ( ∂ r g tt ) ( ∂ r g φφ ) ∂ r g φφ , (2.28)where + / − corresponds to co-rotating/counter-rotating orbits, respectively. The orbitalfrequency is thus ν φ = Ω φ / π . For the especial case of the circular orbits, we have theexpressions E = − g tt + g tφ Ω φ (cid:113) − g tt − g tφ Ω φ − g φφ Ω φ , (2.29) L z = g tφ + g φφ Ω φ (cid:113) − g tt − g tφ Ω φ − g φφ Ω φ , (2.30)for a particle orbiting at the constant radius r . One important parameter to take intoaccount is the innermost stable circular orbit (ISCO). In the Schwarzschild metric this issimply r ISCO = 6 M . The position of ISCO for Kerr- Q metric can be found perturbativelyby using d V eff dr = 0. After some algebra, we obtain r ISCO = − a M + 4 (cid:114) a − M q M. (2.31)This expression reduces to r ISCO = 6 M for the Schwarzschild metric where a = q = 0.Let us suppose that the particle orbit is slightly perturbed along r and θ directions. Wehave r = r (1 + δ r ) and θ = π/ δ θ . Using Eq. (2.26), we find that the equations forperturbations are given by d δ r dt + Ω r δ r = 0 , d δ θ dt + Ω θ δ θ = 0 , (2.32)where the frequencies of the oscillations are [36, 55]Ω r = − g rr ˙ t ∂ V eff ∂r , Ω θ = − g θθ ˙ t ∂ V eff ∂θ . (2.33)– 6 –ccording to the precession model [25], the QPOs are related to the periastron precessionfrequency and nodal precession frequency. The periastron precession frequency ν p and thenodal precession frequency ν n can be obtained as ν p = ν φ − ν r , ν n = ν φ − ν θ . (2.34)Figures 1 and 2 show the orbital, periastron and nodal frequencies for the Kerr- Q metric wherewe have used M/M (cid:12) = 5 . a/M =0.2. We have shown the q = 0 case for comparison.Figure 2 shows the relations of QPO frequencies. q = = / M ν n ( H z ) q = = = =
06 7 8 9 1005001000150020002500 r / M ν ( H z ) Figure 1 . Left : nodal frequency as a function of r/M . Right : ν φ (red) and ν p (blue) as a functionof r/M . We have assumed M/M (cid:12) = 5 . a/M =0.2. q = = ( ν p ( Hz ) ) l n ( ν n ( H z )) q = = ( ν n ( Hz )) l n ( ν ϕ ( H z )) q = = ( ν p ( Hz )) l n ( ν ϕ ( H z )) Figure 2 . Interdependence of three frequencies. – 7 – .2 Rotating Hartle-Thorne metric
The other solution that treats quadrupole to linear order and rotation to second order is theHartle-Thorne metric [49]. This metric is given by ds = − F dt + 1 F dr + G r (cid:34) dθ + sin θ (cid:18) dφ − Jr dt (cid:19) (cid:35) , (2.35)where F = (cid:18) − Mr + 2 J r (cid:19) (cid:20) J M r (cid:18) Mr (cid:19) P ( y ) + 2˜ q Q ( x ) P ( y ) (cid:21) , (2.36) F = (cid:18) − Mr + 2 J r (cid:19) (cid:20) J M r (cid:18) − Mr (cid:19) P ( y ) + 2˜ q Q ( x ) P ( y ) (cid:21) , (2.37) G = 1 − J M r (cid:18) Mr (cid:19) P ( y ) + 2˜ q (cid:34) M (cid:112) r ( r − M ) Q ( x ) − Q ( x ) (cid:35) P ( y ) , (2.38)where x = − r/M , y = cos θ , and ˜ q is defined by˜ q := 58 Q − J /MM = q − J /M M . (2.39)The definitions for the Legendre functions of the second kind in interval x ∈ [1 , ∞ ) are givenby Q ( x ) = − ( x − / (cid:20) x ln (cid:18) x + 1 x − (cid:19) − x − x − (cid:21) , (2.40) Q ( x ) = 32 ( x −
1) ln (cid:18) x + 1 x − (cid:19) − x x − x − . (2.41)It is shown that when the quadrupole moment is given by Q K = J /M , that is q K = a M ,the metric will reduce to the Kerr metric in Boyer-Lindquist coordinates after a coordinatetransformation [47]. q = = q K / M ν n ( H z ) q = = q K q = = q K / M ν ( H z ) Figure 3 . Left : nodal frequency as a function of r/M for the Hartle-Thorne metric.
Right : ν φ (red)and ν p (blue) as a function of r/M . We have assumed M/M (cid:12) = 5 . a/M =0.2. – 8 – = q K q = ( ν p ( Hz )) l n ( ν n ( H z )) q = = q K ( ν n ( Hz )) l n ( ν ϕ ( H z )) q = q K q = ( ν p ( Hz )) l n ( ν ϕ ( H z )) Figure 4 . Interdependence of three frequencies for the Hartle-Thorne metric.
Figure 3 shows the QPOs for the Hartle-Thorne metric where we have used
M/M (cid:12) = 5 . a/M =0.2. We have shown the q K case for comparison with the Kerr metric. Figure 4shows the interdependence of three frequencies, as in Fig. 2. In a similar way to the Kerr- Q metric, the ISCO for the Hartle-Thorne can be derived. The expression for the ISCO reads r ISCO = − . a M − (cid:114) a + 1 . QM + 6 M . (2.42)
In this section our aim is to fit the metrics proposed in the previous sections to investigatethe quadrupole of GRO J1655 −
40. GRO J1655 −
40 is an X-ray binary system where oneof the stars is probably a BH [56]. The measurement of QPOs for this source has beenrealized through RXTE observations [57, 58]. Three QPO frequencies have been measuredusing X-ray timing method [30, 51]. We assume that this compact object is not a KerrBH but a rotating source with a quadruple described by the Kerr- Q metric or the Hartle-Thorne metric. To find the best parameters we use a Bayesian approach. Assuming that thelikelihood is given by L ∼ e − χ / , we use the χ -square defined in Ref. [59]. We also assumethat the emission occurs at r = (cid:15) r ISCO . For each case, we find the value of (cid:15) for which thefit is a good fit. The χ -square takes the following form χ ( a, q, M ) = ( ν C − ν n ) σ + ( ν L − ν p ) σ + ( ν U − ν φ ) σ , (3.1)where ν C , ν L and ν U , as well as their errors denoted by σ i ( i ∈ { C , L , U } ), are provided by theobservations. Let us mention that we should not compare the values of r in different metrics– 9 –e present here, as they have different meaning associated with the underlying geometryand are not physical observables unlike the quadrupoles which are measurable by inertialobservers at infinity.For GRO J1655 −
40 the corresponding frequencies have been measured based on RossiXTEobservations [51] as ν C = 17 . , σ C = 0 . ,ν L = 298 Hz , σ L = 4 Hz ,ν U = 441 Hz , σ U = 2 Hz . (3.2) q = 0.40 +0.320.27 . . . . a / M a/M = 0.38 +0.040.04 . . . . q . . . . . M / M . . . . a/M . . . . . M / M M / M = 5.49 +0.070.07 Figure 5 . Two dimensional marginal posterior for parameters in the Kerr- Q metric. The onedimensional distributions are also shown by the diagonal figures. The contours show 68%, 95% and99% credible intervals. Solid lines represent the values at the maximum of the likelihood given byEq. (3.3). Dashed lines are the 5%, 50%, and 95% percentiles of the distribution. Q metric Let us start with the Kerr- Q metric. Using the observed frequencies in Eq. (3.2), we can findthe maximum of the likelihood L ∼ e − χ / numerically. We also assume that r = 1 . r ISCO – 10 – = 0.24 +0.250.17 . . . . . a / M a/M = 0.37 +0.070.05 . . . . q . . . . . M / M . . . . . a/M . . . . . M / M M / M = 5.41 +0.030.03 Figure 6 . Same as Fig. 5, for model 1. Solid lines represent the values at the maximum of thelikelihood given by Eq. (3.5). as it is a good fit. Therefore, the maximum of likelihood occurs at q = 0 . , M/M (cid:12) = 5 . ,a/M = 0 . . (3.3)Now let us introduce the posterior for the observation. We define the posterior as p ( a/M, M, q | ν i ) ∝ L p ( q ) p ( a/M ) , (3.4)where we have non-informative flat priors for 0 < q < . < a/M < .
8. We alsoassume flat priors for
M/M (cid:12) ∈ (4 ,
7) throughout this work. We use PyMC3 for samplingthe posterior [60].The covariance between variables is illustrated in Fig. 5. The contours show 68%, 95%and 99% credible intervals respectively. The best fits within 68% credibility are q = 0 . +0 . − . , a/M = 0 . +0 . − . and M/M (cid:12) = 5 . +0 . − . . According to the figure it is hard to discriminatethe Kerr metric and the Kerr- Q metric. That is because the specific combination of q and a/M that appears in the χ -square, there is a degeneracy in determining q and a/M . In thisregard, it is a combination of a/M and q that can be measured more precisely.– 11 – = 3.56 +1.111.84 . . . . a / M a/M = 0.39 +0.080.07 . . . . . . . M / M . . . . a/M . . . . M / M M / M = 5.46 +0.010.02 Figure 7 . Same as Fig. 5, for model 2. Solid lines represent the values at the maximum of thelikelihood given by Eq. (3.6).
In this section we construct two models based on the Hartle-Thorne metric. We also proposea novel test of no-hair theorem by directly constraining the coefficient that appears in themetric as we will explain. Let us consider two models based on the Hartle-Thorne metric inthis section. In the first model we treat q as a free parameter and in the second model weassume q ( a, M, β ) = βa M where β is related to the inner structure of the compact object. If β = 1 this is an evidence that the object is a Kerr BH. As before we have the expression for χ -square given in Eq. (3.1) and the observed valuesfor the frequencies and the errors are provided in Eq. (3.2). The likelihood is also given by L ∼ e − χ / . Moreover, we find that the emission at r = 1 . r ISCO is a good fit. This way wefind that the maximum of the likelihood is given by q = 0 . , M/M (cid:12) = 5 . ,a/M = 0 . . (3.5)– 12 –he covariances are illustrated in Fig. 6. The best fit values are q = 0 . +0 . − . , a/M =0 . +0 . − . and M/M (cid:12) = 5 . +0 . − . . Similarly we see that it is difficult to measure q withprecision. It is a combination of q and a/M that can be measured more precisely. Here we propose a new test of no-hair theorem by adapting the Hartle-Thorne metric. Themodel we consider here is a configuration in which q = βa M . If β = 1 we recover Kerr metricto second order in a . In a similar fashion, the maximum of the likelihood is given by β = 4 . , M/M (cid:12) = 5 . ,a/M = 0 . , (3.6)where we have assumed that the emission occurs at r = 1 . r ISCO . The covariances areillustrated in Fig. 7. The best fit values are β = 3 . +1 . − . , a/M = 0 . +0 . − . and M/M (cid:12) =5 . +0 . − . . Because we have βa /M in the expression for χ -square, we see that a combinationof β and a/M is confined with more precision. We see that although the best fit weakly hintstowards some compact object that differs from a Kerr BH, at 3- σ confidence the Kerr solutionlies well in the allowed parameter space. More precise measurements or a combination ofmeasurements could help decrease the uncertainty in parameters. We have provided models to test the no-hair theorem by considering three families of solutionsbased on two metrics that describe a rotating object augmented with a quadrupole. The firstcase study and the first metric is the Kerr- Q metric derived from approximating an exactsolution. We encounter a degeneracy to constrain the spin and the quadrupole. This hindersour ability to measure the quadrupole with more precision. This makes it hard to discriminatethe Kerr or beyond-Kerr nature of the compact object. As a result we find that the KerrBH lies in the 3- σ credible region in the allowed parameter space. The second metric is theHartle-Thorne solution. This metric describes the exterior of rotating compact objects witha quadrupole. As our second study we treat the quaropole as a free parameter. Again, wefind a degeneracy in constraining the spin and the quadrupole and consistency with a KerrBH. In our third case study, we devise a novel test of no-hair theorem by treating thequadrupole as a function of the spin by introducing a parameter β . If the object is a KerrBH, then β = 1. We find that although best fit weakly hints towards some other compactobject rather than the Kerr BH, the Kerr BH lies well in the 3- σ credible interval in theparameter space. Acknowledgments
This work was supported by the National Natural Science Foundation of China (11991053,11721303, 11975027), the Young Elite Scientists Sponsorship Program by the China Asso-ciation for Science and Technology (2018QNRC001), the National SKA Program of China(2020SKA0120300), and the Max Planck Partner Group Program funded by the Max PlanckSociety. – 13 – eferences [1] J. Kormendy and L. C. Ho, Ann. Rev. Astron. Astrophys. (2013), 511-653doi:10.1146/annurev-astro-082708-101811 [arXiv:1304.7762 [astro-ph.CO]].[2] Hansen, R. O. 1974, J. Math. Phys. 15, 46[3] T. Johannsen and D. Psaltis, Astrophys. J. , 11 (2011) doi:10.1088/0004-637X/726/1/11[arXiv:1010.1000 [astro-ph.HE]].[4] A. Tripathi, S. Nampalliwar, A. B. Abdikamalov, D. Ayzenberg, C. Bambi, T. Dauser,J. A. Garcia and A. Marinucci, Astrophys. J. , no.1, 56 (2019)doi:10.3847/1538-4357/ab0e7e [arXiv:1811.08148 [gr-qc]].[5] D. Psaltis, Gen. Rel. Grav. , no.10, 137 (2019) doi:10.1007/s10714-019-2611-5[arXiv:1806.09740 [astro-ph.HE]].[6] H. Liu, M. Zhou and C. Bambi, JCAP , 044 (2018) doi:10.1088/1475-7516/2018/08/044[arXiv:1801.00867 [gr-qc]].[7] T. Johannsen, C. Wang, A. E. Broderick, S. S. Doeleman, V. L. Fish, A. Loeb and D. Psaltis,Phys. Rev. Lett. , no.9, 091101 (2016) doi:10.1103/PhysRevLett.117.091101[arXiv:1608.03593 [astro-ph.HE]].[8] Y. Ni, M. Zhou, A. Cardenas-Avendano, C. Bambi, C. A. R. Herdeiro and E. Radu, JCAP ,049 (2016) doi:10.1088/1475-7516/2016/07/049 [arXiv:1606.04654 [gr-qc]].[9] T. Johannsen, Class. Quant. Grav. , no.12, 124001 (2016)doi:10.1088/0264-9381/33/12/124001 [arXiv:1602.07694 [astro-ph.HE]].[10] T. Johannsen, Class. Quant. Grav. , no.11, 113001 (2016)doi:10.1088/0264-9381/33/11/113001 [arXiv:1512.03818 [astro-ph.GA]].[11] T. Johannsen, Phys. Rev. D , no.12, 124017 (2013) doi:10.1103/PhysRevD.87.124017[arXiv:1304.7786 [gr-qc]].[12] T. Johannsen and D. Psaltis, Astrophys. J. , 57 (2013) doi:10.1088/0004-637X/773/1/57[arXiv:1202.6069 [astro-ph.HE]][13] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], “Observation of GravitationalWaves from a Binary Black Hole Merger”, Phys. Rev. Lett. , no. 6, 061102 (2016)[arXiv:1602.03837 [gr-qc]].[14] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], “GW151226: Observation ofGravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence”, Phys. Rev. Lett. , no. 24, 241103 (2016) [arXiv:1606.04855 [gr-qc]].[15] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], “Binary Black Hole Mergers inthe first Advanced LIGO Observing Run”, Phys. Rev. X , no. 4, 041015 (2016) Erratum:[Phys. Rev. X , no. 3, 039903 (2018)] [arXiv:1606.04856 [gr-qc]].[16] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], “GW170608: Observation of a19-solar-mass Binary Black Hole Coalescence”, Astrophys. J. , no. 2, L35 (2017)[arXiv:1711.05578 [astro-ph.HE]].[17] B. P. Abbott et al. [LIGO Scientific and VIRGO Collaborations], “GW170104: Observation ofa 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2”, Phys. Rev. Lett. , no. 22,221101 (2017) Erratum: [Phys. Rev. Lett. , no. 12, 129901 (2018)] [arXiv:1706.01812[gr-qc]].[18] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], “GW170814: A Three-DetectorObservation of Gravitational Waves from a Binary Black Hole Coalescence”, Phys. Rev. Lett. , no. 14, 141101 (2017) [arXiv:1709.09660 [gr-qc]]. – 14 –
19] M. A. Abramowicz and W. Kluzniak, Astron. Astrophys. (2001), L19doi:10.1051/0004-6361:20010791 [arXiv:astro-ph/0105077 [astro-ph]].[20] D. R. Pasham, T. E. Strohmayer and R. F. Mushotzky, Nature (2014), 74doi:10.1038/nature13710 [arXiv:1501.03180 [astro-ph.HE]].[21] D. R. Pasham, R. A. Remillard, P. C. Fragile, A. Franchini, N. C. Stone, G. Lodato, J. Homan,D. Chakrabarty, F. K. Baganoff and J. F. Steiner, et al.
Science (2019), 531doi:10.1126/science.aar7480 [arXiv:1810.10713 [astro-ph.HE]].[22] T. M. Belloni, A. Sanna and M. Mendez, Mon. Not. Roy. Astron. Soc. (2012) 1701doi:10.1111/j.1365-2966.2012.21634.x [arXiv:1207.2311 [astro-ph.HE]].[23] Samimi, J. et al., 1979. GX339-4: a new black hole candidate. Nature 278, 434-436.doi:10.1038/278434a0.[24] Motch, C., Ricketts, M.J., Page, C.G., Ilovaisky, S.A., Chevalier, C., 1983. SimultaneousX-ray/optical observations of GX339-4 during the May 1981 optically bright state. A&A 119,171-176.[25] L. Stella and M. Vietri, Astrophys. J. Lett. (1998), L59 doi:10.1086/311075[arXiv:astro-ph/9709085 [astro-ph]].[26] L. Stella and M. Vietri, Phys. Rev. Lett. , 17-20 (1999) doi:10.1103/PhysRevLett.82.17[arXiv:astro-ph/9812124 [astro-ph]].[27] L. Stella, M. Vietri and S. Morsink, Astrophys. J. Lett. , L63-L66 (1999)doi:10.1086/312291 [arXiv:astro-ph/9907346 [astro-ph]].[28] C. A. Perez, A. S. Silbergleit, R. V. Wagoner and D. E. Lehr, Astrophys. J. , 589-604(1997) doi:10.1086/303658 [arXiv:astro-ph/9601146 [astro-ph]].[29] A. S. Silbergleit, R. V. Wagoner and M. Ortega-Rodriguez, Astrophys. J. , 335-347 (2001)doi:10.1086/318659 [arXiv:astro-ph/0004114 [astro-ph]].[30] A. Ingram and S. Motta, New Astron. Rev. , 101524 (2019) doi:10.1016/j.newar.2020.101524[arXiv:2001.08758 [astro-ph.HE]].[31] S. E. Motta, A. Rouco-Escorial, E. Kuulkers, T. Mu˜noz-Darias and A. Sanna, Mon. Not. Roy.Astron. Soc. , no. 2, 2311 (2017) doi:10.1093/mnras/stx570 [arXiv:1703.01263[astro-ph.HE]].[32] C. Bambi and S. Nampalliwar, EPL , no. 3, 30006 (2016) doi:10.1209/0295-5075/116/30006[arXiv:1604.02643 [gr-qc]].[33] Z. Stuchl´ık and A. Kotrlov´a, Gen. Rel. Grav. , 1305-1343 (2009)doi:10.1007/s10714-008-0709-2 [arXiv:0812.5066 [astro-ph]].[34] L. Rezzolla, S. Yoshida, T. J. Maccarone and O. Zanotti, Mon. Not. Roy. Astron. Soc. ,L37 (2003) doi:10.1046/j.1365-8711.2003.07018.x [astro-ph/0307487].[35] J. D. Schnittman, J. Homan and J. M. Miller, Astrophys. J. , 420-426 (2006)doi:10.1086/500923 [arXiv:astro-ph/0512595 [astro-ph]].[36] A. Maselli, P. Pani, R. Cotesta, L. Gualtieri, V. Ferrari and L. Stella, Astrophys. J. , no. 1,25 (2017) doi:10.3847/1538-4357/aa72e2 [arXiv:1703.01472 [astro-ph.HE]].[37] N. Franchini, P. Pani, A. Maselli, L. Gualtieri, C. A. R. Herdeiro, E. Radu and V. Ferrari,Phys. Rev. D , no. 12, 124025 (2017) doi:10.1103/PhysRevD.95.124025 [arXiv:1612.00038[astro-ph.HE]].[38] A. G. Suvorov and A. Melatos, Phys. Rev. D , 024004 (2016)doi:10.1103/PhysRevD.93.024004 [arXiv:1512.02291 [gr-qc]]. – 15 –
39] K. Boshkayev, J. Rueda and M. Muccino, Astron. Rep. , no. 6, 441 (2015).doi:10.1134/S1063772915060050[40] A. Maselli, L. Gualtieri, P. Pani, L. Stella and V. Ferrari, Astrophys. J. , no. 2, 115 (2015)doi:10.1088/0004-637X/801/2/115 [arXiv:1412.3473 [astro-ph.HE]].[41] V. Cardoso and P. Pani, Living Rev. Rel. , no. 1, 4 (2019) doi:10.1007/s41114-019-0020-4[arXiv:1904.05363 [gr-qc]].[42] E. Berti, E. Barausse, V. Cardoso, L. Gualtieri, P. Pani, U. Sperhake, L. C. Stein, N. Wex,K. Yagi and T. Baker, et al. Class. Quant. Grav. , 243001 (2015)doi:10.1088/0264-9381/32/24/243001 [arXiv:1501.07274 [gr-qc]].[43] D. Psaltis, Living Rev. Rel. , 9 (2008) doi:10.12942/lrr-2008-9 [arXiv:0806.1531 [astro-ph]].[44] D. Psaltis, J. Phys. Conf. Ser. , 012033 (2009) doi:10.1088/1742-6596/189/1/012033[arXiv:0907.2746 [astro-ph.HE]].[45] K. Jusufi, M. Azreg-A¨ınou, M. Jamil, S. W. Wei, Q. Wu and A. Wang, arXiv:2008.08450[gr-qc].[46] M. Azreg-A¨ınou, Z. Chen, B. Deng, M. Jamil, T. Zhu, Q. Wu and Y. K. Lim, Phys. Rev. D , no. 4, 044028 (2020) doi:10.1103/PhysRevD.102.044028 [arXiv:2004.02602 [gr-qc]].[47] A. Allahyari, H. Firouzjahi and B. Mashhoon, Class. Quant. Grav. , no. 5, 055006 (2020)doi:10.1088/1361-6382/ab6860 [arXiv:1908.10813 [gr-qc]].[48] S. Toktarbay and H. Quevedo, Grav. Cosmol. , 252 (2014) doi:10.1134/S0202289314040136[arXiv:1510.04155 [gr-qc]].[49] J. B. Hartle and K. S. Thorne, “Slowly Rotating Relativistic Stars. II. Models for NeutronStars and Supermassive Stars”, Astrophys. J. , 807-834 (1968).[50] A. Allahyari, H. Firouzjahi and B. Mashhoon, Phys. Rev. D , no. 4, 044005 (2019)doi:10.1103/PhysRevD.99.044005 [arXiv:1812.03376 [gr-qc]].[51] S. E. Motta, T. M. Belloni, L. Stella, T. Mu˜noz-Darias and R. Fender, Mon. Not. Roy. Astron.Soc. , no.3, 2554-2565 (2014) doi:10.1093/mnras/stt2068 [arXiv:1309.3652 [astro-ph.HE]].[52] C. Hoenselaers, W. Kinnersley and B. C. Xanthopoulos, J. Math. Phys. , 2530-2536 (1979)doi:10.1063/1.524058[53] F. Frutos-Alfaro and M. Soffel, Roy. Soc. Open Sci. , 180640 (2018) doi:10.1098/rsos.180640[arXiv:1606.07173 [gr-qc]].[54] B. Toshmatov, D. Malafarina and N. Dadhich, Phys. Rev. D , no.4, 044001 (2019)doi:10.1103/PhysRevD.100.044001 [arXiv:1905.01088 [gr-qc]].[55] A. Merloni, M. Vietri, L. Stella and D. Bini, Mon. Not. Roy. Astron. Soc. , 155 (1999)doi:10.1046/j.1365-8711.1999.02307.x [arXiv:astro-ph/9811198 [astro-ph]].[56] J. A. Orosz and C. D. Bailyn, Astrophys. J. (1997), 876 doi:10.1086/303741[arXiv:astro-ph/9610211 [astro-ph]].[57] T. E. Strohmayer, ApJ , L49–L53 (2001).[58] R. A. Remillard, E. H. Morgan, J. E. McClintock, C. D. Bailyn and J. A. Orosz,[arXiv:astro-ph/9806049 [astro-ph]].[59] C. Bambi, Eur. Phys. J. C , no.4, 162 (2015) doi:10.1140/epjc/s10052-015-3396-7[arXiv:1312.2228 [gr-qc]].[60] J. Salvatier, Thomas V. Wiecki, and Ch. Fonnesbeck, “Probabilistic programming in Pythonusing PyMC3,” PeerJ ComputerScience2, e55 (2016)., no.4, 162 (2015) doi:10.1140/epjc/s10052-015-3396-7[arXiv:1312.2228 [gr-qc]].[60] J. Salvatier, Thomas V. Wiecki, and Ch. Fonnesbeck, “Probabilistic programming in Pythonusing PyMC3,” PeerJ ComputerScience2, e55 (2016).