Testing the Bardeen metric with the black hole candidate in Cygnus X-1

In general, it is very difficult to test the Kerr-nature of an astrophysical black hole candidate, because it is not possible to have independent measurements of both the spin parameter $a_*$ and possible deviations from the Kerr solution. Non-Kerr objects may indeed look like Kerr black holes with different spin. However, it is much more difficult to mimic an extremal Kerr black hole. The black hole candidate in Cygnus X-1 has the features of a near extremal Kerr black hole, and it is therefore a good object to test the Kerr black hole paradigm. The 3$\sigma$-bounds $a_*>0.95$ and $a_*>0.983$ reported in the literature and valid in the Kerr spacetime become, respectively, $a_*>0.78$ and $|g/M|<0.41$, and $a_*>0.89$ and $|g/M|<0.28$ in the Bardeen metric, where $g$ is the Bardeen charge of the black hole.

the BH mass and spin angular momentum. 1 Both the continuumfitting method and the iron line analysis have been extended to non-Kerr spacetimes to test the nature of BH candidates [4,5]. However, in general it is only possible to constrain some combination between the spin and possible deviations from the Kerr solution: the spectrum of non-Kerr objects can be very similar to the one of Kerr BHs with a different spin parameter.
While a number of very exotic metrics can already be ruled out by current observations with these techniques [6], some more theoretically motivated non-Kerr metrics are very difficult to test and the combination of the continuum-fitting method and the iron line analysis cannot solve the degeneracy between the spin and the deformation parameters [7]. The difficulty to constrain deviations from the Kerr metric independently of the measurement of the spin is common to other (potentially more sophisticated) approaches, like the observation of quasi-periodic oscillations [8] and of the X-ray polarization [9]. This problem might be fixed in the future, at least partially, by combining one of the techniques above E-mail address: bambi@fudan.edu.cn. 1 Throughout the Letter, I use units in which G N = c = 1, unless stated otherwise. with the estimate of jet power [10] or the measurement of the BH shadow [11]. However, the observational features of near extremal Kerr black holes are peculiar, and therefore the observation of a BH candidate that looks like a near extremal Kerr BH can constrain the nature of these objects.
While the analysis of the iron line profile is potentially more powerful, the continuum-fitting method is based on more solid physics, the uncertainty on the final measurement is more reliable, and therefore this technique is at present more suitable to derive conservative but robust bounds. It is based on the study of the soft X-ray component of stellar-mass BH candidates, which is interpreted as the thermal spectrum of a geometrically thin and optically thick accretion disk. Among the BH candidates studied with the continuum-fitting method, there are two objects that look like near extremal Kerr BHs: GRS 1915+105 [12] and Cygnus X-1 [13,14]. Studies of the iron line analysis support this conclusion [15] (but see Ref. [16], whose discrepancy is probably due to the improper data state -not high/soft state -and the improper usage in the continuum model in extracting the skewed iron line profile [13]). GRS 1915+105 is a very peculiar source, whose data are difficult to interpret, and the measurement reported in Ref. [12] is based on the assumption that the jet observed in this source is perpendicular to the accretion disk, which is at least questionable. The studies in Refs. [13,14] of Cygnus X-1 are more recent and the results may be thought to be more robust. Under the assumption that the geometry of the spacetime around this object is described by the Kerr metric, the 3σ -bound on the spin parameter of the BH candidate in Cygnus X-1 is found to be a * > 0.95 in [13] and a * > 0.983 in [14]. = r 2 − 2mr + a 2 = 0 has no real roots). The thick red solid line and the thick blue dashed line are the boundaries of the allowed regions for the BH candidate in Cygnus X-1 inferred, respectively, from the 3σ -bound obtained in the Kerr metric a * > 0.95 in Ref. [13] and a * > 0.983 in Ref. [14]. The aim of this Letter is to test the Bardeen metric, which describes the spacetime of a singularity-free BH and can be formally obtained by coupling Einstein's gravity to a non-linear electrodynamics field [17]. In Boyer-Lindquist coordinates, the metric of the rotating solution is equivalent to the Kerr metric with the mass M replaced by the function m, given by [18] where r is the radial coordinate and g is the magnetic charge of the non-linear electrodynamics field (or simply the deformation measuring the deviations from the Kerr metric). As g has the same dimension as M, in analogy with the dimensionless spin parameter a * it is convenient to introduce the dimensionless deformation parameter g * = g/M. Let us note that m → M at large radii. The radius of the event horizon, which is given by the largest root of = r 2 − 2Mr + a 2 = 0 (where a = J /M is the specific spin angular momentum) in the Kerr spacetime, is now the largest root of = r 2 − 2mr + a 2 = 0. Bardeen BHs exist only below a critical spin parameter a c * , which depends on the value of g * and reduces to the Kerr one a c * = 1 for g * = 0, while for a * > a c * there is no horizon and the metric can be thought to describe the gravitational field of a configuration of exotic matter (see Fig. 1). The Bardeen metric can be seen as the prototype of a large class of non-Kerr BH metrics, in which the metric tensor in Boyer-Lindquist coordinates has the same expression of the Kerr one with M replaced by a mass function m(r) that depends only on the radial coordinate r and that reduces to M as r → ∞ [18].
This class of metrics is particularly difficult to test, because the corresponding disk's thermal spectrum and iron line profile are extremely similar to the one corresponding to a Kerr BH with different spin [7].
The standard procedure to test the Bardeen metric with the X-ray data of the BH candidate in Cygnus X-1 would be to repeat the studies of Refs. [13,14] with the Kerr background replaced by the Bardeen one. This would require a detailed analysis to include a large number of astrophysical effects and would be very time consuming. However, we can arrive at the same result with a much faster approach. The key-point is that the thermal spectrum of a thin disk can be used to fit only one parameter of the background geometry. If we assume the Kerr metric, we can infer the BH spin parameter a * . If we relax the Kerr BH hypothesis, we can measure some combination of the spin parameter and of possible deviations from the Kerr solution. As shown in Fig. 2, the disk's thermal spectrum of a Bardeen BH with specific values of a * and g * is practically indistinguishable from the one of a Kerr BH with spinã * ( = a * ). Bardeen BHs with the same g * and spin parameter higher (lower) than a * look like Kerr BHs with spin parameter higher (lower) thanã * . Since here the goal is to test the background metric around the BH candidate in Cygnus X-1, we do not need to repeat the analysis of Refs. [13,14] with the Bardeen background, but we can simply translate the Kerr measurements a * > 0.95 and a * > 0.983 to a bound on a * and g * by comparing the theoretical predictions of disk's thermal spectrum around Kerr and Bardeen BHs.
Calculation method. The calculations of the thermal spectrum of a geometrically thin and optically thick accretion disk have been already extensively discussed in the literature, for both Kerr and non-Kerr spacetimes [2,4]. The theoretical framework is the Novikov-Thorne model [19]. The exact expression of the background metric enters the equations governing the time-averaged radial structure of the disk, the calculation of the propagation of the radiation from the disk to the distant observer, the motion of the particles of gas in the disk (determining the Doppler redshift and blueshift), and the inner edge of the disk, which is thought to be at the ISCO radius in the Novikov-Thorne model and eventually is the most important ingredient to infer the properties of the spacetime around the BH candidate.
The Novikov-Thorne model assumes that the disk is on the equatorial plane and that the disk's gas moves on nearly geodesic circular orbits. The time-averaged energy flux emitted from the surface of the disk is [19] where F (r) is the dimensionless function Here E, L, and Ω are, respectively, the conserved specific energy, the conserved axial-component of the specific angular momentum, and the angular velocity for equatorial circular geodesics; G = −α 2 g rr g φφ is the determinant of the near equatorial plane metric, where α 2 = g 2 tφ /g φφ − g tt is the lapse function; r in is the inner radius of the accretion disk, which is assumed to be the radius of the ISCO. Since the disk is in thermal equilibrium, the emission is blackbody-like and we can define an effective tem- Stefan-Boltzmann constant. The local specific intensity of the radiation emitted by the disk is where T col (r) = f col T eff is the color temperature and f col is the color factor, which takes non-thermal effects into account ( f col ≈ 1.6 in our case). ν e is the photon frequency, h is the Planck constant, k B is the Boltzmann constant, and Υ is a function of the angle between the wavevector of the photon emitted by the disk and the normal of the disk surface, say ξ . The two most common options are Υ = 1 (isotropic emission) and Υ = 1 2 + 3 4 cos ξ (limbdarkened emission). The spectrum can be conveniently written in terms of the photon flux number density as measured by a distant observer: For instance, a Bardeen BH with g * = 0.41 cannot mimic a Kerr BH with a * = 1, but only Kerr BHs with a * 0.95, because the maximum value of the spin parameter of a Bardeen BH with g * = 0.41 is a * ≈ 0.775. See the text for more details.
where I obs , E obs , and ν are, respectively, the specific intensity of the radiation, the photon energy, and the photon frequency measured by the distant observer. I e (ν e )/ν 3 e = I obs (ν obs )/ν 3 follows from the Liouville theorem. dΩ obs = d X dY /D 2 is the element of the solid angle subtended by the image of the disk on the observer's sky, X and Y are the coordinates of the position of the photon on the sky, as seen by the distant observer, while D is the distance of the source. A 1 and A 2 are given by (reintroducing the constants G N and c) . (6) w is the redshift factor and it turns out to be (for the details, see e.g. [4] where λ = k φ /k t is a constant of the motion along the photon path. Doppler boosting, gravitational redshift, and frame dragging are entirely encoded in the redshift factor w, while the effect of light bending enters the integral in Eq. (5), as every photon on the sky is associated with its emission point on the disk (this is done by integrating backward in time null geodesics from the observer to the disk).
In the Kerr spacetime, the model has five free parameters: the BH mass, M, the mass accretion rate,Ṁ, the spin parameter, a * , the disk inclination angle with respect to the line of sight of the distant observer, i, and the distance of the object, D. If M, i, and D can be estimated from independent measurements (e.g. optical observations), one can fit the X-ray data of the thermal spectrum of the disk and infer a * andṀ. 2 In the Bardeen metric, there is a sixth free parameter, g * . However, as discussed above, the spectrum is somehow degenerate with respect a * and g * , in the sense that these two parameters cannot be determined independently, but it is only possible to infer a certain combination of them. In principle, the mass accretion rateṀ can be determined independently, from the low-energy part of the spectrum, whose photons are mainly emitted at large radii, where gravity is almost Newtonian. This is not really true in the case of X-ray data of stellar-mass BH candidates. However, in the case of good measurement it is a good approximation to assume that the determination of a * (or a * and g * ) is not correlated to the one ofṀ.

Results.
We can now compare the thermal spectrum of thin disks around Kerr and Bardeen BHs to translate the bound of the spin parameter of the BH candidate in Cygnus X-1 found in Refs. [13,14] under the assumption of Kerr geometry into an allowed region on the spin parameter-deformation parameter plane of the Bardeen solution. We can make the calculation with the physical parameter of Cygnus X-1 (M = 14.8M , i = 27.1 deg, D = 1.86 kpc, and evenṀ = 1.3 · 10 17 g/s, which is roughly the mean mass accretion rate found from the data analyzed in [13,14]). However, the final result is quite independent of this choice, as in the end we compare Kerr and Bardeen spectra with the same model parameters except the ones providing the geometry of the spacetime. For more exotic metrics, this may not be true. For instance, a different Keplerian velocity and photon propagation may affect, respectively, the Doppler effect and the light bending, so the final constraint may depend on the disk's inclination angle i.
The 3σ -bound a * > 0.95 found in [13] for the Kerr metric becomes the region inside the red solid line of Fig. 1 for the Bardeen background. The constraint is a * > 0.78, |g * | < 0.41.
The 3σ -bound a * > 0.983 found in [14] gives instead the region inside the blue dashed line of Fig. 1 and the bound is a * > 0.89, |g * | < 0.28.
The region of objects without event horizon shown in Fig. 1 can be immediately ruled out for two reasons. First, like in the case of the Kerr metric with g * = 0 [20], the thermal spectrum of disks around horizonless configurations is much harder, due to the sudden drop of the ISCO at smaller radii. Second, even if for g * = 0 horizonless configurations can be created [21], they are expected to be highly unstable, due to the existence of the ergoregion and of stable orbits with negative energy [22].