Relativistic reflection spectra of super-spinning black holes

We construct a relativistic reflection model in a non-Kerr spacetime in which, depending on the value of the deformation parameter of the metric, there are black hole solutions with spin parameter $|a_*|>1$. We apply our model to fit Suzaku data of four supermassive black holes (Ton S180, Ark 120, 1H0419-577, and Swift J0501.9-3239) that have been previously interpreted as near-extremal Kerr black holes. For Ton S180 and 1H0419-577, our measurements are still consistent with the Kerr hypothesis. For Swift J0501.9-3239 and, in particular, Ark 120, the Kerr solution is only recovered at a high confidence level. We discuss our results and possible systematic uncertainties in the model.

We construct a relativistic reflection model in a non-Kerr spacetime in which, depending on the value of the deformation parameter of the metric, there are black hole solutions with spin parameter |a * | > 1. We apply our model to fit Suzaku data of four supermassive black holes (Ton S180, Ark 120, 1H0419-577, and Swift J0501.9-3239) that have been previously interpreted as near-extremal Kerr black holes. For Ton S180 and 1H0419-577, our measurements are still consistent with the Kerr hypothesis. For Swift J0501.9-3239 and, in particular, Ark 120, the Kerr solution is only recovered at a high confidence level. We discuss our results and possible systematic uncertainties in the model.

I. INTRODUCTION
In 4-dimensional Einstein's theory of general relativity, uncharged black holes are described by the Kerr solution [1] and are completely characterized by two parameters, which are associated, respectively, to the mass M and the spin angular momentum J of the compact object. Kerr black holes are subject to the constraint |a * | ≤ 1, where a * = J/M 2 is the dimensionless spin parameter. For |a * | > 1, there is no horizon and the Kerr metric describes the spacetime of a naked singularity.
The spacetime metric around an astrophysical black hole formed from the complete collapse of a progenitor body should be well approximated by the stationary, axisymmetric, and asymptotically-flat Kerr metric, because initial deviations from the Kerr background [2], the presence of nearby stars [3], the gravitational field of accretion disks [4], as well as possible non-vanishing electric charges [5] have normally a very small impact on the spacetime metric and their effects can be completely ignored. All the available black hole observations are consistent with the hypothesis that these objects are the Kerr black holes of Einstein's gravity [6][7][8][9][10][11]. However, there are also a number of theoretical arguments motivating more precise tests that could potentially discover new physics [12][13][14][15][16][17].
Among the electromagnetic methods, X-ray reflection spectroscopy is quite a promising tool to test astrophysical black holes [40][41][42]. In Refs. [43,44], we have presented relxill nk, which is an extension of the relx- * Corresponding author: bambi@fudan.edu.cn ill package [45,46] to parametric black hole spacetimes. In our previous studies, we have mainly considered the Johannsen metric [47], but the model can be easily implemented for any stationary, axisymmetric, and asymptotically-flat black hole spacetime. From the analysis of X-ray data of astrophysical black holes with relxill nk, we can measure the "deformation parameters" of the spacetime and thus constrain possible deviations from the Kerr geometry.
Fast-rotating black holes are the most suitable sources for this kind of tests, because the inner edge of the accretion disk can be very close to the compact object, relativistic features in the reflection spectrum are maximized, and it is thus possible to break the parameter degeneracy and get good measurements of the spacetime metric. For some supermassive black holes, we have obtained remarkably strong constraints on possible deviations from the Kerr metric and spin parameters stuck to the maximum value allowed by the model; see Refs. [48,49] for more details. However, such extreme values of the spin parameter have to be taken with caution. Best fit values at the boundary of the parameter space allowed by the model could indeed be a sympton of the breakdown of the model itself.
In the present work, we further explore the opportunities to test the Kerr hypothesis with such apparently near-extremal Kerr black holes. We consider a subset of the black hole spacetimes proposed in Ref. [50]. In addition to the mass and the spin angular momentum, these spacetimes are characterized by the deformation parameter α. For α = 0, we exactly recover the Kerr solution. For α > 0, there are solutions describing black holes with |a * | > 1 and very small values of the radial coordinate of the event horizon. Implementing such a metric in relxill nk, we fit Suzaku observations of four supermassive black holes (Ton S180, Ark 120, 1H0419-577, and Swift J0501.9-3239) that have been previously interpreted as near-extremal Kerr black holes in [48]. With the new metric, we find that the best-fits of Ton S180 and 1H0419-577 are still consistent with the Kerr hypothesis. On the contrary, the fits of Ark 120 and Swift J0501.9-3239 requires |a * | > 1 and the Kerr solution is only recovered at a higher confidence level. We discuss our findings, the systematic uncertainties in our model, and the choice of astrophysical sources to perform tests of the Kerr metric for future studies.
The paper is organized as follows. In Section II, we review the black hole metric proposed in Ref. [50]. In Section III, we construct a relativistic reflection model for such a metric. In Section IV, we apply our model to analyze Suzaku data of four supermassive black holes: Ton S180, Ark 120, 1H0419-577, and Swift J0501.9-3239. Our results are discussed in Section V. Throughout the paper, we employ units in which G N = c = 1 and the convention of a metric with signature (− + ++).

II. SUPER-SPINNING BLACK HOLES
If we want to test the nature of astrophysical black holes, the most logical method would be compare the predictions of general relativity with those of another theory of gravity in which uncharged black holes are not described by the Kerr solution. In such a case, we should analyze the observational data of a specific source with the Kerr model of general relativity and with the non-Kerr model of the other theory of gravity, and then check if observations prefer one of the two models and can rule out the other one. This is the so-called top-down approach. However, there are two problems. First, there are many theories of gravity, so we should repeat the same test for a large number of theories. Second, usually we do not know the rotating black hole solutions in theories beyond general relativity! This is just a technical problem to solve the corresponding field equations: while it is relatively easy to find spherically symmetric, nonrotating black hole solutions, it is definitively more difficult to find the complete axisymmetric, rotating black hole metrics of a theory of gravity.
The alternative approach, which is often preferred in literature, is the so-called bottom-up method. Now we try to consider an extension of the Kerr metric by adding extra pieces to the Kerr solution. The magnitude of these extra terms is regulated by some "deformation parameters", which are used to quantify possible deviations from the Kerr background. These spacetimes are commonly called parametric black hole spacetimes because they are black hole metrics in which deviations from the Kerr background are parametrized by the deformation parameters. With the spirit of performing a null experiment, we analyze astronomical data with a model assuming such an extended Kerr metric and we try to constrain the value of the deformation parameters to check whether observations require that these parameters vanish, thus confirming the Kerr solution of Einstein's gravity.
In this work, we want to consider a subset of the family of black hole metrics proposed in Ref. [50]. In Boyer-Lindquist coordinates, the line element reads where In what follows, we will only consider the special case α is the deformation parameter of the metric and the Kerr solution is recovered for α = 0. Note that such a form of m 1 and m 2 only alters the strong field region and therefore the metric can pass all Solar System and binary pulsars tests [50]. The metric in Eq. (1) has some interesting properties. In particular, it has very compact and fast-rotating black holes, and matter falling onto similar objects can release more energy than matter falling onto a Kerr black hole [50]. This is quite a relevant property because: i) extremal Kerr black holes are characterized by a very high radiative efficiency and, when we consider deformations of the Kerr metric, we usually obtain black holes with lower radiative efficiency, and ii) observations show that astrophysical black holes can have a very high radiative efficiency, which often helps to get strong constraints on possible deviations from the Kerr solution.
The radial coordinate of the event horizon is given by the largest root of ∆ = 0. For α = 0, we recover the Kerr result R H = M + √ M 2 − a 2 , which ranges from 2M for a non-rotating black hole to M for an extremal black hole with |a * | = 1, and there is no horizon for |a * | > 1. For arbitrary α, we find a richer phenomenology. As shown in the left panel of Fig. 1, for α > 0 R H can be either larger than 2M or smaller than M , depending on the exact values of a * and α. Moreover, the horizon still exists for |a * | > 1. The boundary between spacetimes with black holes and spacetimes with naked singularity is described by Another important quantity is the radial coordinate of the innermost stable circular orbit (ISCO), R ISCO . In the Kerr metric, it monotonically decreases from 9M for an extremal black hole and a counterrotating orbit to 6M for a non-rotating black hole and to M for an extremal black hole and a corotating orbit. As shown in the right panel of Fig. 1, for α > 0 the ISCO radius can be smaller than M .

III. X-RAY REFLECTION SPECTROSCOPY
Accreting black holes can be surrounded by geometrically thin and optically thick disks when the accreting gas has a large angular momentum and the accretion luminosity is between a few percent to about 30% of the Eddington limit of the source. The temperature of the inner part of these accretion disks is in the soft X-ray band for stellar-mass black holes (M ∼ 10 M ) and in the optical/UV band for supermassive black holes (M ∼ 10 5 -10 10 M ). Thermal photons from the disk can inverse Compton scatter off free electrons of the so-called corona, which is a generic name to indicate a hotter (∼ 100 keV) cloud of gas near the black hole. The corona may be represented by the accretion flow plunging from the disk to the black hole, the hot atmosphere of the accretion disk, the base of the jet, etc. The process generates a powerlaw component with an exponential cut-off correlated to the corona temperature. A fraction of this power-law component can illuminate the disk, producing a reflection component. X-ray reflection spectroscopy refers to the analysis of this reflection component.
In the rest-frame of the accreting gas, the reflection spectrum is characterized by some fluorescent emission lines in the soft X-ray band and by the Compton hump at 20-30 keV. The most prominent emission line is often the iron Kα complex, which is at 6.4 keV in the case of neutral or weakly ionized iron and shifts up to 6.97 keV in the case of H-like iron ions (while there is no line in the case of fully ionized iron). The reflection spectrum of the disk far from the source is the result of the combination of the reflection components from different points of the disk, each of them differently shifted by relativistic effects occurring in the strong gravity region near the black hole. For a given accretion disk model and a given spacetime metric, we can compute the reflection spectrum of the whole disk far from the source from the reflection spectrum at the emission point. In the reality, we can construct a model with a reflection spectrum at the emission point, an accretion disk, and a metric, and we can fit the data of an accreting black hole to estimate the values of the model parameters.
In Refs. [43,44], we have presented the XSPECcompatible, public, relativistic reflection model relxill nk, which is the natural extension of the relxill package [45,46] to non-Kerr spacetimes. The model employs the approach of the transfer function proposed by Cunningham [51]. The observed flux (measured, for in-stance, in erg cm −2 s −1 Hz −1 ) of the reflection spectrum is obtained by performing the integral where D is the distance of the source from the detection point, r in and r out are, respectively, the inner and the outer edge of the accretion disk, g = ν o /ν e is the redshift factor, ν o and ν e are, respectively, the photon frequencies at the detection point far from the source and at the emission point in the rest-frame of the accreting gas, r e is the radial coordinate of the emission point, and I e is the specific intensity of the radiation at the emission point in the rest-frame of the accreting gas. g * is the relative redshift factor, which is defined as and ranges from 0 to 1. g max = g max (r e , i) and g min = g min (r e , i) are, respectively, the maximum and the minimum redshift factor for the photons emitted at the radial coordinate r e in the accretion disk and when the viewing angle is i. f = f (g * , r e , i) is the transfer function defined as and |∂ (X, Y ) /∂ (g * , r e ) |is the Jacobian between the Cartesian coordinates X and Y of the image of the disk in the plane of the distant observer and the disk variables g * and r e . More details can be found in Refs. [43,44,51] In Eq. (7), the relativistic effects affecting the reflection spectrum are encodes in the transfer function f (and, at some level, in the radial profile of I e if we consider a specific coronal geometry, which is not our case here). Since the evaluation of the transfer function requires the calculation of a large number of null geodesics, from the plane of the distant observer backwards in time to the emission point on the accretion disk, and these calculations are quite time consuming, it is convenient to tabulate the transfer functions into a FITS (Flexible Image Transport System) file. relxill nk reads the FITS file and calculate the integral in Eq. (7) during the data analysis.
In the FITS file, every transfer function is specified by its values of the black hole spin parameter a * , deformation parameter α, and inclination angle of the disk i, which form a grid 30 × 30 × 22. Transfer functions of generic configurations are obtained by interpolation of the transfer functions of this grid. The grid points are more dense where the ISCO radius changes more rapidly. Fig. 2 shows the distribution of the grid points in the parameter space a * vs α. The grid points for the inclination angle of the disk are evenly distributed in 0 < cos i < 1, as discussed in Ref. [44].

IV. DATA ANALYSIS
In this section, we apply our reflection model with the metric in Eq. (1) to the Suzaku data of four supermassive black holes: Ton S180, Ark 120, 1H0419-577, and Swift J0501.9-3239. The choice of these sources is not accidental, but determined by some remarkable properties. First, these objects were analyzed in Ref. [48], where it was found that they are compatible with near-extremal Kerr black holes. The best-fit value of their spin parameter was found to be 0.998, or close to it, which was the maximum value allowed by the model. Deviations from the Kerr metric, if any, were constrained to be small. Moreover, these four sources present quite a simple spectrum, with no intrinsic absorption complicating the data analysis. Further details can be found in Ref. [48] as well as in Ref. [52], where these data were analyzed for the first time.
The basic details of the Suzaku observations analyzed in our study are reported in Tab. I. The reduction of these data was already discussed in Ref. [48]. In summary, we use the data of the three front-illuminated CCD detectors (XIS0, XIS2 if available 1 , XIS3). We use heasoft v6.24 and CALDB version 20180312 for data reduction. Spectra and response files are combined using the ftool ADDASCASPEC. The combined spectra are rebinned to a minimum of 50 counts in order to ensure the validity of the χ 2 fit statistics. In the data analysis, we ignore the energy range 1.7-2.5 keV because of calibration uncertainties 2 .
tbabs describes the Galactic absorption [54]. The hydrogen column density, n H , is not a free parameter in our fit and is frozen to the value calculated using the tool at 3 .
zpowerlw describes the power-law component from the corona. The photon index Γ and the normalization of the component are free parameters in the fit, while the redshift of the source is frozen to the value reported on the NASA extragalactic database (NED).
relxill nk describes the reflection spectrum from the disk [43,44]. The inclination angle of the disk i, the spin parameter a * , the ionization parameter ξ, and the iron abundance A Fe are always free parameters in the fits. For every source, first we fit the data assuming the Kerr metric (so imposing a * ≤ 0.998 and α = 0) and then for the general case with both a * and α completely free. For the intensity profile, we consider either a simple powerlaw or a broken power-law, and we choose the model that provides the best result. Eventually, for Ton S180 and Ark 120, we employ a broken power-law with outer emissivity index frozen to 3 (lamppost coronal set-up) and therefore we have two free parameters, namely the inner emissivity index q in and the breaking radius R br . For 1H0419-577 and Swift J0501.9-3239, we find that a simple power-law is enough to fit the data, so we have only one free parameter in the fit. The photon index of the radiation illuminating the disk is tied to the valure of the photon index in zpowerlw and the cut-off energy E cut is frozen to 300 keV because it cannot be measured with the XIS data reaching 10 keV.
With the exception of Ton S180, the spectra of these sources also show a non-relativistic reflection component from some cold material far from the strong gravity region. Such a component is fitted with xillver [55], where the values of the parameters are tied to those in relxill nk, with the exception of the ionization parameter, which is frozen to log ξ = 0.
In the data of Ark 120, we also see an emission feature around 6.95 keV (consistent with Fe XXVI) and an absorption feature around 6.1 keV. We fit these features by adding two narrow lines with zgauss.
The results of our fits are reported in Tab. II. Every source has two columns. The first column shows the fit of the Kerr model (a * ≤ 0.998 and α = 0). The second column is for the fit with a * and α free. All the uncertainties are at the 90% confidence level for one relevant parameter (∆χ 2 = 2.71). * indicates that the parameter is frozen in the fit. The spectra of the best-fit model 3 http://www.swift.ac.uk/analysis/nhtot/ with the corresponding components and the data to bestfit model ratios for the case of free a * and α are reported in Fig. 3. Fig. 4 shows the constraints on the spin parameter vs deformation parameter plane for the four sources when both a * and α are free in the fit. The red, green, and blue curves correspond, respectively, to the 68%, 90%, and 99% confidence level limits for two relevant parameters (i.e. ∆χ 2 = 2.30, 4.61, and 9.21, respectively). In the gray region, the spacetimes have no black holes but naked singularities, and are ignored in our analysis. Note that these constraints on a * and α are obtained after marginalizing over all the free parameters in the fit. Since there are several free parameters, the algorithm of XSPEC has some problems to scan the parameter space and find the right minimum of χ 2 at every point of the plane a * vs α. This causes some islands in the contour plots in Fig. 4. If we could better scan the parameter space, the 68%, 90%, and 99% confidence level regions in Fig. 4 should slightly increase in size and these islands should disappear.

V. DISCUSSION AND CONCLUSIONS
In Ref. [48], we tested the Kerr hypothesis with seven supermassive black holes by analyzing some Suzaku observations with relxill nk and constraining the deformation parameters α 13 and α 22 of the Johannsen metric. For all sources, our study found that the spacetime metric around the central compact object was consistent with the Kerr solution, with remarkably strong constraints on the deformation parameters α 13 and α 22 . For several sources, the best-fit value of the spin parameter was stuck at the maximum value of the model, which is 0.998, indicating that the inner edge of the accretion disk of these sources is very close to the compact object, and photons from that region are strongly redshifted. Since such an effect usually gets weaker when we deform the Kerr metric, it is quite automatic that similar sources can provide stringent constraints on the values of most deformation parameters. However, best-fit values stuck at the boundary of the parameter space of a model may also look suspicious, indicating a possible breakdown of the model itself.
Motivated by such a doubt, in the present work we have reanalyzed those suspicious sources with a different parametric black hole spacetime. In particular, we chose a metric in which black holes can have a spin parameter a * exceeding 1 and the radiation from the ISCO radius can be more redshifted than in the Kerr background. Not surprisingly, the best-fit values that were stuck at 0.998 in Ref. [48] moved to a * > 1 for all sources. Our results are summarized in Tab. II, where the last row shows the difference between the minimum of χ 2 between the Kerr black hole model and the model with free deformation parameter.
For Ton S180 and 1H0419-577, the new analysis is For Swift J0501.9-3239, the comparison between the Kerr and non-Kerr model gives ∆χ 2 = 7.03. For Ark 120, the Kerr solution is only recovered if we consider a higher confidence level (∆χ 2 = 20.30), indicating that the non-Kerr model provides a better fit. There are two possible interpretations: i) the supermassive object in Ark 120 is not a Kerr black hole, or ii) systematic uncertainties (broadly defined, not taken into account in our analysis) exceed the statistical uncertainties, thus leading to a wrong measurement of the deformation parameter.
There are surely good reasons to suspect that we are in case ii). The modeling simplifications in relxill nk inevitably lead to a number of systematic uncertainties currently not under control, see Refs. [42,56] for their list and possible impact. As stressed in [42,56], it is ex-tremely important to select the right sources for limiting all these systematic uncertainties. For example, relxill nk employs the Novikov-Thorne model for geometrically thin and optically thick disks and assumes that the inner edge is at the ISCO radius. However, we know that geometrically thin disks with inner edge at the ISCO require that the source is in the soft state with an accretion luminosity between a few percent to about 30% of its Eddington limit, while most supermassive black holes have much higher accretion rates, see Tab. 1 in [57]. Ton S180 and 1H0419-577 seem to accrete at the Eddington or super-Eddington limit. For Swift J0501.9-3239, there is no estimate of its accretion luminosity in Eddington units. The estimated accretion luminosity of Ark 120 is at the level of a few percent, and thus consistent with the thin disk model, so a too high accretion luminosity should not be the explanation of the measured non-vanishing α for this source.
Among the possible systematic uncertainties affecting the measurement of accreting black holes using X-ray reflection spectroscopy, our previous studies showed that FIG. 3. Spectra of the best fit models with the corresponding components (upper panels) and data to best-fit model ratios (lower panels) for the four sources of our study. The total spectra are in black, the power law components from the coronas are in red, the relativistic reflection components from the disks are in blue, the non-relativistic reflection components from distant reflectors are in green, the narrow lines are in cyan. Note that in the case of Ark 120 and Swift J0501.9-3239 the total flux (black) and the relativistic reflection component (blue) almost overlap.
the choice of the model of the intensity profile can be crucial in the final measurement of the deformation parameters of the spacetime [56,58,59]. It is possible that a simple power-law or a broken power-law are not enough to fit the Suzaku data of Ark 120, and this may cause the apparent detection of a non-vanishing α.
In conclusion, our work confirms that it is extremely important to select well-understood sources for testing the Kerr hypothesis and, in general, to get reliable measurements of the parameter of a system. While apparent near-extremal Kerr black holes can provides quite stringent constraints on a number of deformation parameters, it is always dangerous to rely on sources in which some best-fit values are stuck at the boundary of a model.   4. Constraints on the spin parameter a * and the deformation parameter α for the four sources of our study. The red, green, and blue curves correspond, respectively, to the 68%, 90%, and 99% confidence level limits for two relevant parameters (∆χ 2 = 2.30, 4.61, and 9.21, respectively). The gray region is ignored in our study because it is the parameter space with naked singularities.