X-ray reflection spectroscopy with Kaluza-Klein black holes

Kaluza-Klein theory is a popular alternative theory of gravity, with both non-rotating and rotating black hole solutions known. This allows for the possibility that the theory could be observationally tested. We present a model which calculates the reflection spectrum of a black hole accretion disk system, where the black hole is described by a rotating solution of the Kaluza-Klein theory. We also use this model to analyze X-ray data from the stella-mass black hole in GRS 1915+105 and provide constraints on the free parameters of the Kaluza-Klein black holes.


I. INTRODUCTION
General relativity, the standard theory of gravitation today, has been applied to a large range of astrophysical phenomena in our Universe. Over the years, it has undergone a plethora of tests; while largely successful in the weak-field regime [1], its predictions in the strongfield regime have recently become testable in a variety of ways [2][3][4]. During this period, various shortcomings of general relativity (GR hereafter), both from the theoretical (e.g, singularities, difficulty integrating with quantum mechanics, the hierarchy problem) as well as observational point of view (e.g., dark matter, dark energy), have been exposed. Resolutions range from conservative (additional fields, extensions with GR as a limit) to radical (modifications to GR) proposals. In either case, without testable predictions it is impossible to determine if the resolution, conservative or radical, is valid.
Black holes (BHs hereafter) promise to be the strongest probes of gravity in our Universe. Their compactness means gravitational effects, of GR or of alternatives to GR, are strongest in their vicinity. Their simplicity within GR means deviations away from GR, if imprinted on BH solutions, could be easily detected. Finally, their ubiquitousness in nature means there are several potential sources to study. Thus, testing alternative theories of gravity is particularly promising with BHs.
Gravitational waves, imaging of BH shadow, and X-ray spectroscopy are the leading techniques for probing astrophysical BHs. Among these, imaging is not expected to provide very strong constraints on alternative theories [5]. Gravitational wave interferometry is the most promising technique, though in some cases, it is expected to be comparable to X-ray spectroscopy [6]. 1 In the present work, our focus is on the X-ray spectroscopy technique. Specifically, we are interested in the reflection spectrum of BH accretion disks [2,9,10]. For Kerr BHs of GR, relxill is the leading model for analysis of the reflection spectrum [11,12]. Some of us have been involved in generalizing this model to non-Kerr metrics [13,14]. The model, relxill nk, has been applied to X-ray observations of several astrophysical BHs to place constraints on deviations away from the Kerr solution [15][16][17][18][19][20][21][22][23][24][25][26][27][28]. A public version of the model is available at [29,30]. Another interesting phenomenon that can potentially be used to probe strong-field gravity is the gyroscope precession frequency. The gravitational field of the BH causes the rotational axis of a gyroscope to precess, and the precession frequency carries unique signatures of the BH metric. Such precession frequencies have been calculated in several non-Kerr spacetimes [31][32][33][34].
One of the most interesting alternative theory of gravity proposals is the Kaluza-Klein theory. With purely classical origins back in 1919, the theory has been interpreted in a quantum mechanical as well as a string theory framework. It is a five-dimensional theory, with a compact fifth dimension. The basic ingredients include three kinds of fields: gravity, electromagnetism and a scalar field. Tests of the theory include looking for signatures in the Large Hadron Collider [35], but such tests have not been very successful yet. Among astrophysical tests, in [36] the authors have analyzed how equations of motion change in Kaluza-Klein cosmology which may affect motions of galaxies. Shadows of Kaluza-Klein BHs have also been analyzed in [37]. Gravitational waves are not expected to provide good constraints in the near future [38] (See also [39]). In [40], some of us study the precession of a gyroscope in the vicinity of a Kaluza-Klein BH. We can therefore ask the question: can the predictions of Kaluza-Klein theory be tested using X-ray spectroscopy? This paper presents our efforts to answer this question.
BHs in Kaluza-Klein theory have been derived in various limits [41]. Non-rotating spherically symmet-ric BHs were derived in [42][43][44]. Larsen (among others) found rotating BH solutions in five and four dimensions [45][46][47]. BHs with squashed horizon were calculated in [48,49]. Six and higher dimension versions have also been found [50]. Since astrophysical BHs are mostly rotating, and X-ray spectroscopy is most suited for rapidly rotating BHs, we shall focus on rotating BHs. Rotating Kaluza-Klein BHs typically have four free parameters: mass, spin and the electric and magnetic charges. We have implemented this BH metric in the relxill nk framework, and used data from an X-ray binary to get constraints on the free parameters.
The paper is organized as follows: in Sec. II, we review the Kaluza-Klein theory and the BH metric. In Sec. III we review the theory of X-ray spectroscopy, the relxill nk framework, and describe the numerical method we used to implement the Kaluza-Klein BH metric in relxill nk. In Sec. IV, the new model is applied to Xray observations of an X-ray binary. Conclusions follow in Sec. V.

II. THE METRIC
We will follow the notation developed in [40] for the metric. The simplest Kaluza-Klein theory involves three fields: gravity, the dilaton and the gauge field. The action in the Einstein frame is [51]: where √ −g is the determinant of the four-dimensional metric tensor, R the Ricci scalar, F αβ the gauge field which can be identified with the electromagnetic field, and σ a dilaton scalar field. κ is a constant and is equal to √ 16πG.
Although we will not derive the BH solution here, it is interesting to point out some features of the solution generating techniques. Standard methods of solving the field equations can be used to derive the non-rotating solutions. Rotating solutions on the other hand have been obtained in the following ways: for slow rotation, [52] solve the complete field equations perturbatively, following [53]; others [41,45] boost the Kerr metric along a line to get a five-dimensional rotating BH solution. The metric looks like this: where the one-forms are given by and The solution admits four free parameters, viz. m, a, p, q, which are related to the physical mass M , the angular momentum J and the electric (Q) and magnetic charge (P ) respectively. The relations are given as: The fifth dimension can be compactified and this results in a four-dimensional BH metric [40]: Here we have used dimensionless version of the free parameters, defined as α ≡ a/M, b ≡ p/m, c ≡ q/m, and x ≡ r/M . Moreover, we can relate the free parameter m and the physical mass M using Eq. 4 and obtain Note that the spin parameter α is not always the same as the dimensionless spin parameter of the Kerr metric. Only when the electric and magnetic charges are zero, and the Kaluza-Klein metric reduces to the Kerr metric, does α equal the a * parameter of the Kerr solution.
We now discuss some properties of this solution. It admits two horizons, viz.
or, in terms of the dimensionless quantities, and the determinant is equal to ρ 2 sin 2 θ. The nonrotating class of solutions (α = 0) was obtained in [44]. For b = c, the non-rotating solution reduces to the Reissner-Nordström solution of GR. The Kerr solution is recovered when b = c = 2. However, when magnetic charge is zero, the metric does not reduce to the Kerr-Newman solution. Here, since we are interested in BHs that parametrically deviate from the Kerr BHs, we will consider solutions with α = 0, b = 0, c = 0. Since our data analysis models allow only one variable deformation parameter, we choose to set b = c, and thus retain one free deformation parameter, which we will henceforth denote as b.
During the analysis, it is important to ensure the spacetime does not have pathologies. Requiring that the metric structure is preserved everywhere outside the horizon leads to bounds on the free parameters. We will determine these bounds now. First, following the definition of Q 2 and P 2 in Eq. 4, we have the conditions q ≥ 2m, p ≥ 2m, or b ≥ 2, c ≥ 2. Using these and Eq. 9 we arrive at a bound on α: For upper bounds on b, c, we first note that we have set b = c. Then, using Eq. 9 and Eq. 10 gives us Thus, III. X-RAY REFLECTION SPECTROSCOPY

A. Theory
The standard astrophysical system we consider is a stationary BH surrounded by an accretion disk. The disk could be generated by a companion (as in stellar-mass X-ray binaries) or galactic material (as around active galactic nuclei). The typical model for a BH-accretion disk system is the disk-corona model. Fig. 1 shows a cartoon of this model. The BH is assumed to be surrounded by an optically thick and geometrically thin disk in the equatorial plane [54], with its inner edge at some radius r in , bounded by the innermost stable circular orbit (ISCO hereafter), and the outer edge at a radius r out . The system also includes a "corona". The corona is a cloud of hot plasma (effective temperature of the order of 100 keV) whose morphology is not understood very well. The radiation spectrum includes a power-law component, produced by inverse Compton scattering of photons from the disk by the corona, a thermal continuum of blackbody radiation from the particles in the disk, and a reflection component, produced when the upscattered photons return to the disk and are reflected after reprocessing inside the disk. Our focus in the present work is on the reflection component. It is a sensitive feature and is affected not just by the BH (driving the photon from the point of emission to an observer) but also by the structure and composition of the disk, as well as the corona.

B. The RELXILL NK model
Reflection models therefore include parameters from all aspects of the disk-corona model. relxill nk [13,14,29,30] is a suite of reflection models built for XSPEC that includes a large class of BH-disk-corona models. To illustrate, we describe the eponymous model, relxill nk. Tab. I lists the parameters of the base relxill nk model as well their default values. These parameters account for the different aspects of the system. The disk's emissivity profile is modeled as a simple or broken power law as follows: The disk is assumed to be infinitesimally thin, lying in the equatorial plane and composed of particles moving in quasi-geodesic circular orbits. Thus only two disk structure parameters r in and r out are needed, to account for the inner and the outer radius of the disk respectively. The elemental constitution of the disk is assumed to follow solar abundances, except iron, which is modeled with A Fe , which is the ratio of iron content in the disk and the iron content in the sun. The ionization of the disk is accounted with log ξ (where ξ is in units of erg cm/s), which ranges from 0 (neutral) to 4.7 (highly ionized). The corona is modeled with a power law, whose power law index is given by Γ and the high energy cut-off by E cut . The latter is an observational feature. The R f parameter controls the relative contributions of the coronal and the reflection spectra, and is defined as the ratio of intensity emitted towards the disk and that escaping to infinity. The spacetime is modeled using three param-eters: the BH spin α, the deformation parameter type, which can be used to switch between different deformation parameters, and the value of the deformation parameter. Notably, the BH mass is not a parameter since the reflection spectrum (unlike the thermal spectrum) does not explicity depend on the BH mass. The observer's viewing angle is modeled with i and finally the strength of the spectrum is accounted with the norm N .

C. Numerical method
The output of the relxill nk model includes the reflection spectrum at the observer. Mathematically, this is given as Here I o is the specific intensity (for instance, in units of erg s −1 cm −2 str −1 Hz −1 ) as detected by an observer. X and Y are the Cartesian coordinates of the image of the disk in the plane of the distant observer, and dΩ = dXdY /D 2 is the element of the solid angle subtended by the image of the disk in the observer's sky. I o can be related to the specific intensity at the point of emission via the Liouville's theorem: I o = g 3 I e , where g = ν o /ν e is the redshift factor, ν o is the photon frequency in the observer's frame at the point of detection, and ν e is the photon frequency in the emitter's rest frame at the point of emission. The integration element dΩ which is presented in terms of variables on the observer plane can also be recast using the redshift factor and the transfer function [55], where the latter is defined as follows: Here, r e is the radial coordinate at the point of emission on the disk and g * is the normalized redshift factor, defined as where g max = g max (r e , i) and g min = g min (r e , i) are, respectively, the maximum and the minimum values of the redshift factor g at a fixed r e and for a fixed viewing angle of the observer. The flux can now be rewritten as where D is the distance of the observer from the source and ϑ e is the photon's direction relative to the disk at the point of emission. The r e -integral ranges from the inner to the outer edge of the disk, and the g * -integral ranges from 0 to 1.
Given the transfer function, the reflection spectrum can be readily calculated using Eq. 16. But it is computationally expensive to calculate the transfer function by tracing photons and using Eq. 14 every time the flux needs to be calculated, therefore relxill nk uses an interpolation scheme to calculate the transfer function for any {g * , r e , i} using a FITS (Flexible Image Transport System) table which stores the transfer functions for some {g * , r e , i}. The procedure to create such a table is explained in detail in [13,14]. Here we give a brief overview. The three physical parameters spin α, deformation parameter b and the observer's viewing angle i, are discretized in a 30 × 30 × 22 grid, respectively. Note that the grid spacing in each dimension is not necessarily uniform, e.g., the grid becomes denser as α increases, since the ISCO radius changes faster with increasing α. This scheme enables sufficient resolution during interpolation while maintaining a reasonable table size. The b dimension of the grid depends on the type of deformation parameter under consideration. For the present study, our deformation parameter b is bounded between 2 and 2/|α| (Eq. 12). We additionally bound b to be below 10. The final b − α grid is shown in Fig. 2. Note that due to numerical complications, in some cases the bounds on b are more conservative. At each grid point (i.e., each α, b, and i value in the FITS table), the accretion disk is discretized in 100 emission radii r e and 40 equally spaced g * values. 2 The emission radii grid ranges from the ISCO to 1000M , and is non-uniform, with higher density near the ISCO. Photons are traced backwards in time from the observer plane to the disk, using the ray-tracing scheme described in [13,14]. An adaptive algorithm fine-tunes the initial location on the observer plane so that the photon, when back-traced, lands at specific r e 's. For each such "central" photon, the code calculates the redshift, emission angle, etc. Moreover, four photons closely spaced in the observer plane are launched to calculate the Jacobian and subsequently the transfer function, using Eq. 14. For each r e , about 100 such redshifts, emission angles and transfer functions are calculated, which are then interpolated to get these quantities on the 40 equally spaced values of g * , which is stored in the FITS table.

IV. DATA ANALYSIS
In this section, we present our analysis of an X-ray observation using the relxill nk model described above.

A. Review
We chose the source GRS 1915+105 for this analysis. GRS 1915+105 (or V1487 Aquilae) is a low mass X-ray binary lying 8.6 kiloparsecs away [56]. It features one of the most massive stellar BHs known in our Galaxy. Since its last outburst in 1992, it has been a persistent source of X-rays. In previous work, we have looked at a NuS-TAR and a Suzaku observation of this source. In [26], we used relxill nk to analyze a 2012 NuSTAR observation. This observation was difficult to fit and resulted in inconsistent values of the deformation parameter. In a follow up work [27], we used relxill nk to analyze a 2012 Suzaku observation, which required fewer components and resulted in a fit consistent with the Kerr metric. Note that fits to the NuSTAR observation required a thermal component, suggesting a hotter disk, unlike the Suzaku observation where no thermal component was required, suggesting a colder disk. Since the relxill nk model is based on xillver which assumes a cold disk, the results of the fits with the Suzaku observation can be expected to be more reliable.
A qualitative picture emerged from previous analyses of GRS 1915+105 thus: the Suzaku observation can be fitted well with the base relxill nk model, the emissivity profile is a broken power law with very high emissivity index in the inner parts of the disk and very small in the outer parts (suggesting a ring-like corona above the accretion disk [57,58]), the spin is high (∼ 0.99), the inclination is high (∼ 60−70 deg) and the spacetime metric is very close to the Kerr metric. Recently, a version of relxill nk developed for thick disks was also used to analyze the Suzaku observation [59], which found that the thick disk version of the relxill nk model provides only a marginally better fit than the base relxill nk model, which assumes infinitesimal thickness.

B. Observations and data reduction
Suzaku observed GRS 1915+105 for 117 kiloseconds on May 7, 2007 (Obs ID 402071010). During this observation, two XIS units were turned off (to preserve telemetry) and a third unit was running in the timing mode, therefore we used data from XIS1 and HXD/PIN instruments only.
The data reduction for this observation has been described in [26,27]. We use the same reduced data in the analysis here. In particular, for the XIS1 camera a net exposure time of 28.94 ks (in the 3 × 3 editing mode) and for the HXD/PIN a net exposure time of 53.00 ks was achieved. For the analysis, we used the 2.3 keV (since after absorption, there are insufficient photons at low energies for fitting) to 10 keV (to avoid calibration issues near the Si K edge) energy band for XIS1 data and 12.0-55.0 keV energy band for HXD/PIN data following [60].

C. Modelling and results
In our analysis, we employed XSPEC 12.10.1f. Since this observation has been analyzed before with relxill nk, it was natural to guess that the best-fit model combination found previously would also work here. We thus fit the observation with the following model: Model: tbabs*relxill nk, where tbabs describes the galactic absorption [61] and we keep the galactic column density free. The coronal and reflection spectrum are modeled with relxill nk. (The thermal spectrum does not feature in this observation, as shown in [60].) The disk emissivity profile is modeled with a broken power law. The disk inner edge lies at the ISCO, a standard assumption considering the source during the observation was in a state considered equivalent to the high-soft state and the Eddington scaled accretion luminosity is 20% [60,62,63], and the outer edge at 400M . The results of the fit are shown in Tab. II. The reduced χ 2 is close to 1, indicating statistical agreement between model and data. The best-fit model and the data to model ratio is presented in Fig. 3, in the top and the bottom panels respectively. There are no outstanding features that appear unresolved, thus we can be confident that the model fits the data satisfactorily. We can compare the best-fit parameter values obtained here with their values in other analyses. The emissivity profile, for example, follows previous results with a high q in , nearly zero q out and the break occurring near 6M . The result can be explained within a ring-like corona above the accretion disk [57,58]. The spin and inclination are high, as found before. The iron abundance is below solar, and the cut off energy is quite low. Of course, our main interest here is the deformation parameter. We find that 2 ≤ b ≤ 2.08, (17) within 90% uncertainty. We can use Eq. 4 to translate this into a constraint on the electric and magnetic charges respectively (Note that we have chosen b = c): Since spin and deformation parameter are generally degenerate, we also show a contour plot of α vs. b in Fig. 4. The red, green and blue curves are for 68%, 90% and 99% confidence, respectively. The degeneracy between spin and b is evident here.

V. CONCLUSION
Alternative theories of gravity have recently become observationally testable in the strong field regime. Among other techniques. X-ray spectroscopy provides one of the strongest constraints on deviations from Einstein's theory. One particularly interesting example of alternatives to general relativity is the Kaluza-Klein theory. In this paper, we use one class of rotating BH solutions of Kaluza-Klein theory described in [40]. We describe the astrophysical system assumed in X-ray spectroscopy analyses and an XSPEC model that calculates the reflection spectrum of this system. The XSPEC model , relxill nk, is then used to analyze a Suzaku observation of the stellar-mass BH system GRS 1915+105. We review the specific source, describe the specific observation, and use our model to fit the data. We find that the observation is consistent with the Kerr metric, but allows for some deviation away from it. It will be interesting to test for signatures of such deviations in other astrophysical sources in future.