Towards precision tests of general relativity with black hole X-ray reflection spectroscopy
Ashutosh Tripathi, Sourabh Nampalliwar, Askar B. Abdikamalov, Dimitry Ayzenberg, Cosimo Bambi, Thomas Dauser, Javier A. Garcia, Andrea Marinucci
TTowards precision tests of general relativity with black hole X-ray reflectionspectroscopy
Ashutosh Tripathi, Sourabh Nampalliwar, Askar B. Abdikamalov, Dimitry Ayzenberg, Cosimo Bambi, ∗ Thomas Dauser, Javier A. Garc´ıa,
4, 3 and Andrea Marinucci Center for Field Theory and Particle Physics and Department of Physics, Fudan University, 200438 Shanghai, China Theoretical Astrophysics, Eberhard-Karls Universit¨at T¨ubingen, 72076 T¨ubingen, Germany Remeis Observatory & ECAP, Universit¨at Erlangen-N¨urnberg, 96049 Bamberg, Germany Cahill Center for Astronomy and Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA Dipartimento di Matematica e Fisica, Universit´a degli Studi Roma Tre, 00146 Roma, Italy
Astrophysical black hole systems are the ideal laboratories for testing Einstein’s theory of gravityin the strong field regime. We have recently developed a framework which uses the reflectionspectrum of black hole systems to perform precision tests of general relativity by testing the Kerrblack hole hypothesis. In this paper, we analyze
XMM-Newton and
NuSTAR observations of thesupermassive black hole in the Seyfert 1 galaxy MCG–06–30–15 with our disk reflection model.We consider the Johannsen metric with the deformation parameters α and α , which quantifydeviations from the Kerr metric. For α = 0, we obtain the black hole spin 0 . < a ∗ < . − . < α < .
15. For α = 0, we obtain 0 . < a ∗ < .
987 and − . < α < .
05. TheKerr solution is recovered for α = α = 0. Thus, our results include the Kerr solution withinstatistical uncertainties. Systematic uncertainties are difficult to account for, and we discuss someissues in this regard. I. INTRODUCTION
Einstein’s gravity has been extensively tested in theweak field regime, its theoretical predictions being largelyconfirmed by experiments in the Solar System and ra-dio observations of binary pulsars [1]. The strong fieldregime, on the other hand, is still largely unexplored.There are many alternative and modified theories of grav-ity that have the same predictions as Einstein’s gravityfor weak fields and present deviations only when gravitybecomes strong. Astrophysical black holes give us an op-portunity to test the predictions of Einstein’s gravity inthe strong field regime [2–6].In 4-dimensional Einstein’s gravity, the only station-ary and asymptotically flat vacuum black hole solution,which is regular on and outside the event horizon, is theKerr metric [7–9]. The spacetime around astrophysicalblack holes is thought to be well approximated by thissolution. Testing the Kerr black hole hypothesis with as-trophysical black holes is thus a test of Einstein’s gravityin the strong field regime, and can be seen as the counter-part of Solar System experiments aimed at verifying theSchwarzschild solution in order to test Einstein’s gravityin the weak field regime [10–18].In this work, we study the X-ray spectrum of the su-permassive black hole in MCG–06–30–15 with the reflec-tion model relxill nk [19] as a step in our programto test the Kerr black hole hypothesis from the reflec-tion spectrum of the disk of accreting black holes [20–24].MCG–06–30–15 is a very bright Seyfert 1 galaxy and ithas been observed for many years by different X-ray mis-sions. It is the source in which a relativistically blurred ∗ Corresponding author: [email protected] iron K α line was clearly detected for the first time [25],and is thus one of the natural candidates for tests of Ein-stein’s gravity using X-ray reflection spectroscopy. Weanalyze simultaneous observations of XMM-Newton [26]and
NuSTAR [27], which provide both high energy reso-lution at the iron line (with
XMM-Newton ) and a broadenergy band (with
NuSTAR ).The contents of the paper are as follows. In Section II,we briefly review the parameterized metric employed inour test and our previous results. In Sections III and IVwe present, respectively, our data reduction and analysis.Section V is devoted to the discussion of our results andthe conclusions. Throughout the paper, we adopt theconvention G N = c = 1 and a metric with signature( − + ++). II. THE REFLECTION MODEL
RELXILL NK relxill nk [19] is the natural extension of relx-ill [28, 29] to non-Kerr spacetimes. It describes thedisk’s reflection spectrum of an accreting black hole [30].Here we employ the Johannsen metric. In Boyer-Lindquist-like coordinates, the line element reads [31] ds = − ˜Σ (cid:0) ∆ − a A sin θ (cid:1) B dt + ˜Σ∆ A dr + ˜Σ dθ − a (cid:2)(cid:0) r + a (cid:1) A A − ∆ (cid:3) ˜Σ sin θB dtdφ + (cid:104)(cid:0) r + a (cid:1) A − a ∆ sin θ (cid:105) ˜Σ sin θB dφ (1)where M is the black hole mass, a = J/M , J is the black a r X i v : . [ g r- q c ] A p r hole spin angular momentum, ˜Σ = Σ + f , andΣ = r + a cos θ , ∆ = r − M r + a ,B = (cid:0) r + a (cid:1) A − a A sin θ . (2)The functions A , A , A , and f are A = 1 + ∞ (cid:88) n =3 α n (cid:18) Mr (cid:19) n , A = 1 + ∞ (cid:88) n =2 α n (cid:18) Mr (cid:19) n ,A = 1 + ∞ (cid:88) n =2 α n (cid:18) Mr (cid:19) n , f = ∞ (cid:88) n =3 (cid:15) n M n r n − . (3)The “deformation parameters” { α n } , { α n } , { α n } , and { (cid:15) n } are used to quantify possible deviations from theKerr background. In what follows, we restrict our atten-tion to the deformation parameters α and α , sincethese two have the strongest impact on the reflectionspectrum [19]. In our analysis, we leave either one of α and α free, setting the other to zero. All otherdeformation parameters are set identically to zero. Inorder to avoid spacetimes with pathological properties,we require | a ∗ | ≤
1, where a ∗ = a/M = J/M is thedimensionless spin parameter, and [21] − (cid:16) (cid:112) − a ∗ (cid:17) < α < (cid:16) (cid:112) − a ∗ (cid:17) a ∗ ,α > − (cid:16) (cid:112) − a ∗ (cid:17) . (4)From the analysis of the reflection spectrum of as-trophysical black holes with relxill nk we can con-strain the deformation parameters α and α and checkwhether they are consistent with zero, as required bythe Kerr black hole hypothesis. In Ref. [20], we ana-lyzed XMM-Newton , NuSTAR , and
Swift data of thesupermassive black hole in 1H0707–495 and we got thefirst constraint of α (see [32] for the same constraintswith an updated version of relxill nk ). In Refs. [21]and [24], we analyzed Suzaku data of, respectively, thesupermassive black hole in Ark 564 and Mrk 335, andwe constrained the deformation parameters α and α .In Refs. [22] and [23], we tested the Kerr nature of thestellar-mass black holes in GX 339–4 and GS 1354–645,respectively.For these five sources, three supermassive black holesand two stellar mass black holes, we have found that thevalue of the deformation parameters is consistent withzero at least within a 90% confidence level (and usuallywithin 68% confidence level). The most stringent con-straints have been obtained from GS 1354–645, wherethe bounds on a ∗ , α , and α are (99% of confidencelevel for two relevant parameters) a ∗ > . − . < α < .
16 (for α = 0) , (5) a ∗ > . − . < α < .
42 (for α = 0) . (6)Our results are thus consistent with the Kerr black holehypothesis, as expected. However, these results were not Mission Observation ID Exposure (ks)
NuSTAR
XMM-Newton obvious a priori considering the possible systematic ef-fects of our model, which are not fully under control.This, in turn, might be interpreted as the fact that thesystematic uncertainties are subdominant for the currentprecision of our tests.
III. OBSERVATIONS AND DATA REDUCTION
MCG–06–30–15 is a very bright Seyfert 1 galaxy atredshift z = 0 . α line, soit is quite a natural candidate for our tests of the Kerrmetric using X-ray reflection spectroscopy. However, thesource is very variable, which requires some attention inthe data analysis. NuSTAR and
XMM-Newton observed MCG–06–30–15simultaneously starting on 29 January 2013 for a totaltime of ∼
315 ks and ∼
360 ks, respectively. Tab. I showsthe observation ID and their exposure time. A study ofthese data was reported in [41].
NuSTAR is comprised of two co-aligned telescopeswith focal plane modules (FPMA and FPMB) [42]. Thelevel 1 data products are analyzed using NuSTAR DataAnalysis Software (NUSTARDAS). The downloaded rawdata are converted into event files (level 2 products) us-ing the HEASOFT task NUPIPELINE and using the lat-est calibration data files taken from NuSTAR calibrationdatabase (CALDB) version 20180312. The size of thesource region is taken to be 70 arcsec centered at thesource and that of the background is 100 arcsec takenfrom same CCD. The final products (light curves, spec-tra) are extracted using the event files and region files byrunning the NUPRODUCTS task. Spectra are rebinnedto 70 counts per bin in order to apply χ statistics.For XMM-Newton , observations from three consecu-tive revolutions are taken with the two EPIC cameras Pnand MOS1/2 operating in medium filter and small win-dow mode [43]. Here, we are only using Pn data owingto their better quality [44]. The MOS data are not usedin our analysis because they suffered from high pile-up.SAS version 16.0.0 is used to convert the raw data intoevent files. These event files are then combined into asingle fits file using ftool FMERGE. Good time intervals(GTIs) are generated using TABTIGEN and then used . . c oun t s k e V − c m − Energy (keV)
FIG. 1. Source (the data in the upper part of the figure) and background (the data with the stars in the lower part of thefigure) spectra for EPIC-Pn, FPMA, and FPMB for all the four flux states considered in this work. The data are divided bythe response effective area of each particular channel. in filtering the event files. For source events, we take acircular region of 40 arcsec centered at the source. Forbackground, we take a 50 arcsec region. After backscal-ing, response files are produced. Finally, in order to applythe χ statistics, spectra are rebinned in order to over-sample the instrumental resolution by at least a factor of3 and have 50 counts in each background-subtracted bin.Source and background spectra of each instrument areshown in Fig. 1.As the source is highly variable, it is important to usesimultaneous data so that variability is properly takeninto account. We use ftool mgtime to find the commonGTIs of the two telescopes. IV. SPECTRAL ANALYSIS
MCG–06–30–15 is highly variable in the X-ray band.This could be due to passing clouds near the black holealong our line of sight and/or variations of the coro-nal geometry, as both phenomena can have a timescaleshorter than our observations. To take such a sourcevariability into account, we have arranged our data infour groups according to the flux state of the source(low flux state, medium flux state, high flux state, andvery-high flux state). Since we have data from three in- struments (
XMM-Newton , NuSTAR /FPMA, and
NuS-TAR /FPMB), in the end we have to deal with 12 datasets. The data are divided into four flux states such thatspectral data counts will be similar in each flux stateas shown in Fig. 2. Luminosities and fluxes in the en-ergy range 0.5-10 keV for
XMM-Newton and 3-80 keVfor
NuSTAR for every data set are shown in Tab. II.Note that our grouping scheme is different from that em-ployed in Ref. [41], which was based on the hardness ofthe source.To combine the
XMM-Newton and
NuSTAR data, weset the constant of
XMM-Newton to 1 and we leave theconstants of
NuSTAR /FPMA and
NuSTAR /FPMB free.After the fit, we check that the ratio between the con-stants of
NuSTAR /FPMA and
NuSTAR /FPMB is be-tween 0.95 and 1.05. Tab. III shows the values of theseconstants for every flux state.As discussed in the appendix of Ref. [41], in the
XMM-Newton data we see a spurious Gaussian around 2 keV.This is interpreted as an effect of the golden edge inthe response file due to mis-calibration in the long-termcharge transfer inefficiency (CTI), i.e. how photon ener-gies are reconstructed after detection. We solve this issueby simply ignoring the energy range 1.5-2.5 keV in the
XMM-Newton /EPIC-Pn data. Such a region is not cru-cial for testing the Kerr metric and therefore its omission
Luminosity (10 erg s − ) Flux (10 − erg cm − s − )Group 1 2 3 4 1 2 3 4 XMM-Newton
NuSTAR /FPMA 1.01 1.24 1.50 2.03 0.76 0.93 1.12 1.53
NuSTAR /FPMB 1.04 1.27 1.52 2.07 0.78 0.95 1.14 1.55TABLE II. Average luminosity (assuming z = 0 . XMM-Newton and 3-80 keV for
NuSTAR for every flux state and instrument. Group 1 is for the low flux state, group 2 is for themedium flux state, group 3 is for the high flux state, and group 4 is for the very-high flux state.FIG. 2.
NuSTAR /FPMA,
NuSTAR /FPMB and
XMM-Newton /EPIC-Pn light curves. The three dashed horizontallines separate the four different flux states.Group 1 2 3 4
XMM-Newton
NuSTAR /FPMA 1.060 1.044 1.043 1.053
NuSTAR /FPMB 1.089 1.065 1.058 1.072TABLE III. Cross calibration constants between
XMM-Newton and
NuSTAR . The constant of
XMM-Newton isfrozen to 1. is not so important for the final result. We cannot addan ad hoc
Gaussian to fit this feature because this wouldalso modify the way in which the warm absorbers/ionizedreflectors reproduce the data.We first try to fit the data of the low flux state with
FIG. 3. Data to best-fit model ratio for the model tbabs × cutoffpl for the low flux state. We can clearly see thereflection features of the spectrum: broad iron line around6 keV, Compton hump around 20 keV, and soft excess below1 keV. Red crosses are used for XMM-Newton , green crossesfor
NuSTAR /FPMA, and blue crosses for
NuSTAR /FPMB. a simple power law to identify the spectral features.The notable features above 3 keV are the iron K α linearound 6.4 keV and the Compton hump at 20-30 keV(see Fig. 3) [48, 49]. Below 3 keV, there are featuresfrom complex ionized absorbers [50, 51]. In [46], the au-thors studied the low energy spectrum of this source andfound that fitting requires two warm absorbers and oneneutral absorber; this is the choice extensively adoptedin the literature [38, 41]. In order to fit the spectrum,we employ the model consisting of the following compo-nents: non-relativistic reflection from distant cold mate-rial, relativistic reflection from the ionized accretion diskand power law for primary emission. We use xillver for the cold reflection [28], relxill nk for the blurredreflection [19], and cutoffpl for the power law emis-sion with free cut-off energy. A narrow emission line anda narrow absorption line are also required. The combi-nation of the above-mentioned models is convolved withtwo ionized absorbers, one dusty absorber, and Galac-tic absorption as mentioned in the literature [38, 41, 46].Tab. IV shows the improvement of the fit as we add newcomponents to the model.The final XSPEC model is tbabs × warmabs × Model χ ν χ /ν tbabs × cutoffpl . In model 1, we add dustyabs to model 0. Inmodel 2, 3, and 4, we add, respectively, one, two, and three warmabs to model 1. In model 5, we add relxill nk tomodel 3. In model 6, we add xillver to model 5. In model 7and 8, we add one and two zgauss , respectively, to model 6. warmabs × dustyabs × (cutoffpl + relxill nk+ xillver + zgauss + zgauss) . tbabs describesthe Galactic absorption and we set the column density N H = 3 . · cm − [45]. warmabs and warmabs are two ionized absorbers and their tables are generatedwith xstar v 2.41. dustyabs is a neutral absorber whichmodifies the soft X-ray band due to the presence of dustaround the source [46]. cutoffpl is a power law withan exponential cut-off and describes the direct radiationfrom the Comptonized corona. relxill nk is our disk’sreflection model for the Johannsen spacetime [19], wherethe reflection fraction parameter is set to −
1, so there isno power-law from the corona because we prefer to use cutoffpl . xillver is the reflection spectrum from someionized non-relativistic matter at larger distance [47]. Af-ter fitting the data with all the above mentioned model FIG. 4. Spectrum of the best-fit of model a of the low fluxstate (red) and its components: power law component (blue),relativistic reflection component (black), non-relativistic re-flection component (magenta), and emission line (green). components, there are features at low energies that canbe fit with gaussian profiles. One of the two zgauss describes a narrow oxygen line around 0.81 keV and theother one describes a narrow absorption at 1.22 keV. Thelatter can be interpreted in terms of blueshifted oxy-gen absorption due to the presence of relativistic out-flows [52]. The spectrum of the best-fit model with itscomponents for the low flux state is shown in Fig 4.We have two models: model a in which α is free and α = 0, and model b where α = 0 and α can vary.Thus, we test for one non-zero deformation parameterat a time. The best-fit values are reported in Tab. Vfor both model a and model b . The estimated error isthe 90% confidence interval for one parameter of interest(∆ χ =2.71). Fig. 5 shows the quality of our fits with theresiduals for model a (for model b we obtain very similarresults).The column densities and the ionization parameters ofthe two warm absorbers are allowed to vary from differ-ent flux states. The neutral iron density in dustyabs isinstead kept constant: it describes the absorption of thedust surrounding the source and its iron density shouldnot change much among different flux states; see [35] formore details about dust absorption in MCG–06–30–15.The photon index Γ and the energy cut-off E cut to de-scribe the spectrum of the corona are allowed to varywith the flux state because the geometry of the coronacan change over the observational timescale.For the disk’s reflection spectrum described by relx-ill nk , we start with an emissivity profile described bya broken power-law and the inner emissivity index q in ,the outer emissivity index q out , and the breaking radius R br all free and allowed to vary with the flux state. How-ever, we find that q out is always consistent with 3, as wecould expect in the case of a lamppost corona. For ex-ample, for model a we get 2 . +0 . − . , 2 . +0 . − . , 3 . +0 . − . ,and 2 . +0 . − . for the four flux states. We thus repeatthe fit freezing q out to 3. The inclination angle of theaccretion disk, the black hole spin, the deformation pa-rameters, and the iron abundances are clearly constantover the observation period. The ionization parameter ξ is allowed to vary because it is affected by the geometryof the corona, which may change at different flux states.The normalization also varies among different flux states.We freeze the reflection fraction of relxill nk to be − cutoffpl andthe cut-off energy is left free because it can be estimatedfrom the NuSTAR data.In xillver , the parameters are tied to those in relx-ill nk , with the exception of the ionization and ironabundance. For the ionization parameter, we set log ξ (cid:48) =0, as the non-relativistic reflection component is thoughtto be produced far away the black hole, in the outer partof the accretion disk or the molecular torus. The ironabundance is fixed at solar value as the distant cold re-flector likely has a low iron abundance. The normaliza-tion is tied between different flux states as the distant Model a b
Group 1 2 3 4 1 2 3 4 tbabs N H / cm − . (cid:63) . (cid:63) warmabs N H 1 / cm − . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . log ξ . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . warmabs N H 2 / cm − . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . log ξ . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . − . . +0 . − . . +0 . − . dustyabs log (cid:0) N Fe / cm − (cid:1) . +0 . − . . +0 . − . cutoffpl Γ 1 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . E cut [keV] 198 +11 − +20 − +22 − +116 − +80 − +71 − +82 − +86 − N cutoffpl (10 − ) 8 . +0 . − . . +0 . − . . +0 . − . . +1 . − . . +0 . − . . +2 . − . . +1 . − . . +0 . − . relxill nk q in . +1 . − . . +0 . − . . +0 . − . . +0 . − . ∼ . +2 . − . . +0 . − . . +0 . − . q out (cid:63) (cid:63) R br [ M ] 2 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . i [deg] 31 . +1 . − . . +2 . − . a ∗ . +0 . − . . +0 . − . α . +0 . − . (cid:63) α (cid:63) . +0 . − . z . (cid:63) . (cid:63) log ξ . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . A Fe . +0 . − . . +0 . − . N relxill nk (10 − ) 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . xillver log ξ (cid:48) (cid:63) (cid:63) N xillver (10 − ) 0 . +0 . − . . +0 . − . zgauss E line [keV] 0 . +0 . − . . +0 . − . zgauss E line [keV] 1 . +0 . − . . +0 . − . χ /dof 3029 . / . . / . a ( α free and α = 0) and model b ( α = 0 and α free). Theionization parameter ξ is in units erg cm s − . The reported uncertainties correspond to the 90% confidence level for onerelevant parameter. (cid:63) indicates that the parameter is frozen. See the text for more details. reflector is not expected to vary much. V. DISCUSSION AND CONCLUSIONS
The primary aim of this work is put constraints onpossible deviations from the Kerr solution. For α =0, our constraints on the black hole spin parameter a ∗ and the Johannsen deformation parameter α are (99% confidence level for two relevant parameters)0 . < a ∗ < . , − . < α < . . (7)When we assume α = 0, we find (still 99% confidencelevel for two relevant parameters)0 . < a ∗ < . , − . < α < . . (8)In both cases, the value of the deformation parameteris consistent with 0, which is the value required by theKerr solution and predicted by Einstein’s gravity. The FIG. 5. Best-fit model and standard deviations for model a . The top left panel is for the low flux state, the top right panel isfor the medium flux state, the bottom left panel is for the high flux state, and the bottom right panel is for the very-high fluxstate. Red crosses are used for XMM-Newton , green crosses for
NuSTAR /FPMA, and blue crosses for
NuSTAR /FPMB. -0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 α a * -0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 0.88 0.9 0.92 0.94 0.96 0.98 α a * FIG. 6. Constraints on the spin parameter a ∗ and on the Johannsen deformation parameter α (left panel) and α (rightpanel). The red, green, and blue lines indicate, respectively, the 68%, 90%, and 99% confidence level contours for two relevantparameters. The thick black horizontal line marks the Kerr solution. confidence level contours a ∗ vs α and a ∗ vs α areshown in Fig. 6.The best-fit values of α and α are very close to 0, soit is relatively straightforward to compare the results ob-tained here with those from previous studies. In general,we can say that the best-fit values of the model parame-ters are consistent with the estimates found in previous analyses in the Kerr background, and in particular withthose found in [41] analyzing the same XMM-Newton and
NuSTAR observations. For the model parameters thatare constant over different flux states, our measurementsare consistent with those in [41]. The estimate of the spinparameter and of the inclination angle of the accretiondisk agree with those in [41]. All previous studies find -0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 4 5 6 7 8 9 α q -0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 4 5 6 7 8 9 α q -0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 4 5 6 7 8 9 α q -0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 4 5 6 7 8 9 α q FIG. 7. Impact on deformation parameter α by emissivity profile of different flux intervals. q , q , q , and q are theinner emissivity indices for low, medium, high and very high flux state, respectively. The red, green, and blue lines indicate,respectively, the 68%, 90%, and 99% confidence level contours for two relevant parameters. The thick black horizontal linemarks the Kerr solution. for this source an iron abundance higher than the Solariron abundance [37]. The iron abundance found in [41] issomewhat lower than ours, but the difference can be eas-ily explained by the different analysis method. For themodel parameters that vary over different flux states, adirect comparison with Ref. [41] is not possible becausethe grouping scheme is different from ours. We just no-tice that here we find that the ionization parameter of therelativistic reflection component nicely increases with theluminosity, as is expected. A. Constraints on the Kerr metric
As in our previous studies, our results are consistentwith the Kerr black hole hypothesis. The constraint on α obtained in the present work from MCG–06–30–15 iscomparable to that obtained from the stellar-mass blackhole in GS 1354–645 [23], while the constraint on α isslightly weaker, see Eqs. (5) and (6).On one hand, the fact that we mostly recover the Kerrsolution is expected: since Einstein’s gravity has alreadysuccessfully passed a large number of observational tests,it is likely that astrophysical black holes are at least verysimilar to, if not exactly, the Kerr black holes of Ein-stein’s gravity. On the other hand, since our model has anumber of simplifications that inevitably introduce manysystematic uncertainties, these confirmations are not atall obvious. It could be that our non-Kerr parameterswere able to absorb some of these systematic uncertain-ties. We are thus tempted to argue that the fact wealways recover Kerr can be interpreted as the systematicuncertainties from the model being currently subdomi-nant relative to the uncertainties due to the quality ofthe data.Model simplifications are both in the description ofthe accretion disk and in the calculations of the reflec-tion spectrum. The disk is assumed to be infinitesimallythin, on the equatorial plane of the black hole, and itsinner edge is set at the innermost stable circular orbit(ISCO). In the reality, the thickness of the disk is finiteand increases with the mass accretion rate. A prelimi-nary study on the impact of the disk thickness on thereflection spectrum has been reported in [53]. The in-ner edge of the disk is thought to be at the ISCO radiuswhen the accretion luminosity is between 5% and 30% ofthe Eddington limit [54, 55], while for higher luminositiesit may move to a smaller radius [56]. For supermassiveblack holes, it is typically difficult to get reliable esti-mates of the accretion luminosity, because of the largeuncertainties in the estimates of their mass and distancefrom us. In the case of MCG–06–30–15, the Edding-ton scaled accretion luminosity has been estimated to be0 . ± .
13 [57], so deviations from the thin disk modelcan be expected even if they may be moderate.There are also a number of simplifications in the cal- culation of the reflection spectrum. Our model currentlyassumes a fixed electron density in the disk, a constantdisk density over height and radius, a single ionizationparameter for the whole disk, etc. At the moment, asystematic study on the impact of these simplificationson the measurements of the spin and the deformationparameters is lacking, but work is underway.The emissivity profile is usually thought to be a cru-cial ingredient and source of systematic uncertainties. Tocheck its impact on the estimate of the deformation pa-rameters, in [23] we showed that incorrect modeling ofthe emissivity profile leads to non-vanishing deformationparameters. The fact that we always recover the Kerrmetric when we fit the emissivity index suggests that thequality of our data is good enough to permit us to esti-mate both the deformation parameter and the emissivityindex, as an accidental compensation leading to recov-ery of the Kerr metric sounds unlikely. However, if theemissivity profile is so important for the estimate of thedeformation parameter, as suggested in [23], we shouldexpect that a power law or a broken power law may notbe adequate in the case of high quality data, hopefullyavailable with the next generation of X-ray missions [58].To further explore the role of the emissivity profile onthe estimate of the deformation parameters, we have plot-ted the constraints on the plane q in vs α for every fluxstate of the observations of MCG–6–30–15 studied in thiswork. Fig. 7 shows the 68%, 90%, and 99% confidencelevel contours for two relevant parameters, where q , q , q , and q are, respectively, the inner emissivity indicesfor low, medium, high and very high flux state. Theseplots do not show any particular correlation betweenthe q in and α , confirming that the spectral analysisof the source can separately determine these two quanti-ties. Such a conclusion cannot, in general, be extended toany deformation parameter, but at least it seems to holdfor α and α . Because of the current uncertainties inthe corona geometry, and therefore in the exact shapeof the emissivity profile, the non-observation of a corre-lation between the emissivity index and the deformationparameters can partially limit the systematic uncertaintyin the estimate of the deformation parameters due to theuncertainty in the correct emissivity profile. Acknowledgments –
A.T. thanks Laura Brennemanfor useful discussions on MCG–06–30–15. This workwas supported by the National Natural Science Foun-dation of China (NSFC), Grant No. U1531117, and Fu-dan University, Grant No. IDH1512060. A.T. also ac-knowledges support from the China Scholarship Coun-cil (CSC), Grant No. 2016GXZR89. S.N. acknowledgessupport from the Excellence Initiative at Eberhard-KarlsUniversit¨at T¨ubingen. A.B.A. also acknowledges the sup-port from the Shanghai Government Scholarship (SGS).J.A.G. acknowledges support from the Alexander vonHumboldt Foundation.0 [1] C. M. Will, Living Rev. Rel. , 4 (2014)[arXiv:1403.7377 [gr-qc]].[2] C. Bambi, Rev. Mod. Phys. , 025001 (2017)[arXiv:1509.03884 [gr-qc]].[3] C. Bambi, J. Jiang and J. F. Steiner, Class. Quant. Grav. , 064001 (2016) [arXiv:1511.07587 [gr-qc]].[4] K. Yagi and L. C. Stein, Class. Quant. Grav. , 054001(2016) [arXiv:1602.02413 [gr-qc]].[5] V. Cardoso and L. Gualtieri, Class. Quant. 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