Iron K α line of Kerr black holes with Proca hair
Menglei Zhou, Cosimo Bambi, Carlos A.R. Herdeiro, Eugen Radu
IIron K α line of Kerr black holes with Proca hair Menglei Zhou, Cosimo Bambi,
1, 2, ∗ Carlos A. R. Herdeiro, and Eugen Radu Center for Field Theory and Particle Physics and Department of Physics, Fudan University, 200433 Shanghai, China Theoretical Astrophysics, Eberhard-Karls Universit¨at T¨ubingen, 72076 T¨ubingen, Germany Departamento de F`ısica da Universidade de Aveiro and Center for Research and Developmentin Mathematics and Applications (CIDMA), Campus de Santiago, 3810-183 Aveiro, Portugal (Dated: October 18, 2018)We continue our study on the capabilities of present and future X-ray missions to test the natureof astrophysical black hole candidates via X-ray reflection spectroscopy and distinguish Kerr blackholes from other solutions of 4-dimensional Einstein’s gravity in the presence of a matter field. Herewe investigate the case of Kerr black holes with Proca hair [1]. The analysis of a sample of theseconfigurations suggests that even extremely hairy black holes can mimic the iron line profile of thestandard Kerr black holes, and, at least for the configurations of our study, we find that currentX-ray missions cannot distinguish these objects from Kerr black holes. This contrasts with ourprevious findings for the case of Kerr black holes with scalar (rather than Proca) hair [2], eventhough such comparison may be biased by the limited sample. Future X-ray missions can detectthe presence of Proca hair, but a theoretical knowledge of the expected intensity profile (currentlymissing) can be crucial to obtain strong constraints.
I. INTRODUCTION
In Einstein’s gravity, vacuum black holes (BHs) aredescribed by the Kerr solution and are completely char-acterized by two parameters, associated, respectively, tothe mass M and the spin angular momentum J of theobject. This “uniqueness theorem” holds under specificassumptions [3–5] (for a review, see e.g. Ref. [6]). Besidesvacuum (no matter energy-momentum tensor is allowed),the spacetime must have 3+1 dimensions and be station-ary, asymptotically flat, regular on and outside the BHevent horizon. The fact that BHs under such conditionscan be completely specified by only two parameters, andnot more, is often given the name of “no-hair theorem”.This should be understood as the absence of other degreesof freedom ( e.g. multipolar structure) independent fromthe two aforementioned global charges, both of which areassociated to Gauss laws and measurable at infinity.On the observational side, there is today a consider-able body of evidence for the existence of BHs in ourGalaxy and in the Universe. Stellar-mass BHs in X-raybinary systems are compact objects with a mass in therange 5-20 M (cid:12) , which is too high to be that of compactrelativistic stars for any reasonable matter equation ofstate [7, 8]. Supermassive BHs in galactic nuclei are tooheavy, compact, and old to be clusters of non-luminousbodies, because the cluster lifetime due to evaporationand physical collision would be too short, thus makingtheir existence in the present Universe highly unlikely [9].The non-detection of thermal radiation from the putativesurface of all these objects is consistent with the fact thatthey are BHs with an event horizon and there is no sur-face [10, 11]. The gravitational waves recently detected ∗ Corresponding author: [email protected] by the LIGO experiment are also consistent with the sig-nal expected from the coalescence of stellar-mass BHs inEinstein’s gravity [12–14].The spacetime metric of astrophysical BHs formedfrom the complete gravitational collapse of very massivestars is thought to be well described by the Kerr solu-tion. Initial deviations from the Kerr geometry are ex-pected to be quickly radiated away with the emission ofgravitational waves immediately after the creation of theevent horizon [15]. Due to the difference between the pro-ton and electron masses, the BH electric charge may benon-vanishing, but its equilibrium value is reached soon,because of the highly ionized host environment of theseobjects, and its impact on the background metric is com-pletely negligible [16]. The presence of an accretion diskcan be ignored, because the density of the disk is low andits mass is several orders of magnitude smaller than themass of the BH [17, 18].In this context, within General Relativity, a crucialassumption is that the Kerr metric is still the only phys-ical BH solution even in the presence of matter [19]. Itturns out that even though the spirit of the uniquenesstheorems can be extended, under assumptions, to differ-ent types of matter contents, yielding different no-hairtheorems (see e.g. [20] for a review), it is possible toobtain BHs with “hair” in Einstein’s gravity. That is,BH solutions which are not fully described by parame-ters measurable at infinity and associated to Gauss laws(see [20–22] for recent reviews). Whereas many of theseexamples have matter contents that violate the energyconditions, hairy BHs with a simple matter content andobeying all energy conditions have been recently discov-ered. These solutions are characterized by having syn-chronized hair , in which the matter field has a phase an-gular velocity that matches the angular velocity of therotating horizon. This synchronous rotation mechanism a r X i v : . [ g r- q c ] M a y implies a vanishing matter field flux through the hori-zon and provides an equilibrium between the matter fieldand the horizon, preventing the collapse of the formerinto the latter. Examples of solutions with synchronizedhair include Kerr BHs with scalar hair [23–25] and Procahair [1]. The key-ingredient to circumvent classical no-hair theorems is that the bosonic field is complex andtime periodic. The existence of hairy black hole solu-tions involving massive scalar fields has been proven rig-orously in [26]. These solutions interpolate continuouslybetween vacuum Kerr BHs and a solitonic limit (scalarboson stars [27] and Proca stars [28], respectively). Assuch one expects their phenomenological properties tovary continuously between those of a standard Kerr BHand those of the corresponding solitons, which can po-tentially be very non-Kerr like.There are two main approaches to test the nature ofastrophysical BH candidates: with electromagnetic radi-ation [29] and with gravitational waves [30]. In this pa-per, we continue our explorative study to understand thecapabilities of present and future X-ray missions to testBH solutions in 4D Einstein’s gravity in the presence ofmatter using X-ray reflection spectroscopy, the so-callediron line method [31–34]. We extend previous work onKerr BHs with scalar hair [2], boson stars [35], and Procastars [36]. Other phenomenological studies of BHs withsynchronized hair include their shadows [37–40], Quasi-Periodic-Oscillations [41] and some brief analyses of theirquadrupoles and orbital frequency at the Innermost Sta-ble Circular Orbit [23, 24].The content of the present paper is as follows. InSections II and III, we briefly review, respectively, KerrBHs with Proca hair and X-ray reflection spectroscopy.In Section IV, we present our simulations with theXIS instrument on board of the Suzaku X-ray mission(XIS/Suzaku) and with the LAD detector expected tobe on board of eXTP (LAD/eXTP) and our best-fits. InSection V, we discuss our results. Summary and conclu-sions are in Section VI. Throughout the paper we employnatural units in which c = G N = (cid:126) = 1 and a metric withsignature ( − + ++). II. KERR BLACK HOLES WITH PROCA HAIR
Kerr BHs with Proca hair are solutions to Einstein’sgravity minimally coupled to a complex Proca field [1]and may represent the final product of the non-linear evo-lution of super-radiant instability [42]. The correspond-ing action is S = (cid:90) d x √− g (cid:18) R π − F αβ ¯ F αβ − µ A α ¯ A α (cid:19) , (1)where A is the complex Proca potential 1-form with fieldstrength F = d A and complex conjugates ¯ A and ¯ F , re-spectively; µ is the Proca field mass. BH solutions to this model can be found using thefollowing metric ansatz [1] ds = − e F ( r,θ ) N ( r ) dt + e F ( r,θ ) (cid:18) dr N ( r ) + r dθ (cid:19) + e F ( r,θ ) r sin θ [ dϕ − W ( r, θ ) dt ] , (2)where N ( r ) ≡ − r H r , (3)and r H is the horizon radial coordinate; the Proca po-tential ansatz is [1] A = e i ( mϕ − wt ) [ iV ( r, θ ) dt + H ( r, θ ) dr + H ( r, θ ) dθ + iH ( r, θ ) sin θdϕ ] . (4)All of the above functions are real functions. Solutionsexist under the synchronization condition w = m Ω H , (5)where Ω H = W ( r = r H , θ ) is the ( θ -independent) horizonangular velocity.In Fig. 1 we exhibit the domain of existence for KerrBHs with Proca hair in an ADM mass vs. vector fre-quency diagram, both in units of the Proca field mass µ .This domain of existence is for solutions with m = 1 andalso with the smallest number of nodes of the Proca fieldpotential. The two plotted boundaries for this domainof existence correspond to the solitonic limit (rotatingProca stars – solid red line) and the Kerr limit (blue dot-ted line), corresponding to the subset of vacuum Kerrsolutions that can support a Proca stationary cloud ( i.e a test field configuration on the Kerr background) with m = 1. The BH solid line corresponds to the set of ex-tremal Kerr BHs in such a diagram, translating w intoΩ H via (5). Vacuum Kerr BHs exist below the blacksolid line; thus the vacuum Kerr limit of this family ofhairy BHs (blue dotted line) is a 1-dimensional subset ofthe two dimensional parameter space. In (a subset of)the white region just above the black solid line around ω/µ ∼ .
9, there may still exist hairy BHs, in particularclose to the border of the blue shaded area. However,this region is particularly challenging numerically, andno accurate solutions could be obtained.In Fig. 1 there are three highlighted configurations,dubbed III, IV and V (to match their numbering in [1]),corresponding to the solutions that shall be analyzed be-low. These can be briefly described as a BH with “littlehair” (only a relatively small fraction of the total massand angular momentum is stored in the Proca field andthe metric is thus fairly close to Kerr solution – config-uration III), a “very hairy” BH (most of its mass andangular momentum are stored in the Proca field – con-figuration IV) and an “extremely hairy” BH (the massand the angular momentum are almost fully in the Procafield – configuration V). Their main physical propertiesare summarized in table II (adapted from [1]).
Config. w/µ r H /µ µM ADM µ J ADM a ADM ∗ µM H µ J H µM ( P ) µ J ( P ) III 0.98 0.25 0.365 0.128 0.961 0.354 0.117 0.011 0.011IV 0.86 0.09 0.915 0.732 0.874 0.164 0.070 0.751 0.663V 0.79 0.06 1.173 1.079 0.784 0.035 0.006 1.138 1.073TABLE I. Configurations to be examined in this paper. Here M ADM and J ADM denote the ADM mass and angular momentum, M H and J H denote the horizon mass and angular momentum, and M ( P ) and J ( P ) denote the energy and angular momentumstored in the Proca field. The dimensionless spin parameter is a ADM ∗ = J ADM /M ADM . Details on how these quantities arecomputed may be found in [1]. Observe that for configurations III, IV and V, the percentages of energy and angular momentumstored inside the horizon are, respectively (cid:39) (97%, 91%), (18%, 9.5%) and (3.0%, 0.6%). This justifies designating the threeBHs as having little hair, being very hairy and being extremely hairy. The data is publicly available in [43]. M µ w/ µ m=1 vacuum Kerr black holesKerr black holes with Proca hair Proca StarsIIIIVV
FIG. 1. Domain of existence of Kerr BHs with Proca hair, inan ADM mass vs. vector field frequency diagram in units ofthe Proca field mass µ . III. X-RAY REFLECTION SPECTRUM
X-ray reflection spectroscopy is a technique that canbe employed to probe the spacetime metric in the stronggravity region around a BH by studying the reflectionspectrum of its accretion disk. Within the disk-coronamodel [44, 45], a BH is surrounded by a geometricallythin and optically thick accretion disk. The disk emitslike a blackbody locally and as a multi-color blackbodywhen integrated radially. The “corona” is a hot, usu-ally optically thin, cloud of gas close to the BH, but itsexact geometry is currently unknown. For instance, itmay be the base of the jet, a sort of atmosphere abovethe accretion disk, or some accretion flow between thedisk and the BH. Due to inverse Compton scattering ofthe thermal photons from the disk off the hot electronsin the corona, the latter becomes an X-ray source witha power-law spectrum. Some X-ray photons from thecorona illuminate the disk, producing a reflection com-ponent with some emission lines. The most prominentfeature is usually the iron K α line, which is at about6.4 keV for neutral and weakly ionized atoms but it canshift up to 6.97 keV in the case of H-like iron ions.The accretion disk can be described by the Novikov- Thorne model [46], which is the standard set-up for ge-ometrically thin and optically thick accretion disk. Thedisk is in the equatorial plane perpendicular to the spinof the compact object. The particles of the disk follownearly geodesic circular orbits in the equatorial plane.The inner edge of the disk is located at the innermoststable circular orbit (ISCO). When the particles of thegas reach the ISCO, they quickly plunge onto the centralobject, so we can neglect the radiation emitted inside theISCO.The iron K α line is a very narrow emission line in therest frame of the emitting gas, while the line in the re-flection spectrum of BHs is broad and skewed, as a resultof special and general relativistic effects occurring in thestrong gravity region around the compact object. If prop-erly understood and in the presence of high quality data,the study of the disk reflection spectrum can be a pow-erful tool to probe the geometry around BHs and checkwhether the spacetime metric is indeed well described bythe Kerr solution [47–52].As an explorative work, for the sake of simplicity herewe do not consider the full reflection spectrum and werestrict our attention to the iron K α line only, which isthe most prominent feature and the most sensitive oneto the strong gravity region. The calculations of the ironline profile as detected by an instrument in the flat far-away region have been extensively discussed in the liter-ature, and in our case we employ the code described inRefs. [53, 54] and readapted to treat numerical metricsin [55]. This is essentially a ray-tracing code. We firephotons from a grid in the image plane of the distantobserver to the accretion disk and we find the emissionpoints. We calculate the redshift factor from the photonconstants of motion and the gas velocity at the emissionpoint. We repeat the calculations for every point in thegrid and we then integrate over the disk image to get thetotal spectrum as seen by the distant observer.The shape of the iron line depends on the backgroundmetric (which determines all the relativistic effects: grav-itational redshift, Doppler boosting, light bending), theinclination angle of the disk with respect to the line ofsight of the distant observer, the emissivity profile of theaccretion disk. The latter is a weak point in the cal-culations, because it should depend on the geometry ofthe corona, which is currently unknown. It is commonto parametrize the intensity profile for a generic coronageometry with a power-law (the intensity is proportionalto 1 /r q , where r is the emission radius and q is the emis-sivity index) or with a broken power-law (namely 1 /r q for r < r br and 1 /r q for r > r br , where q and q are,respectively, the inner and the outer emissivity indicesand r br is the breaking radius).Fig. 2 shows the iron line profiles for the configura-tions III, IV, and V of Kerr BHs with Proca hair. In bothpanels, the inclination angle of the disk with respect tothe line of sight of the distant observer is i = 45 ◦ andthe rest-frame energy of the line is at 6.4 keV. In the left panel, it was assumed the intensity profile 1 /r . In theright panel, we assumed the lamppost-inspired intensityprofile I ∝ h ( h + r ) / . (6)It corresponds to the intensity profile expected in theNewtonian theory (no light bending) in the case thecorona is a point-like source at the height h from thecentral object and along its spin axis [56]. h of orderof a few gravitational radii is what we can expect if, forinstance, the corona is the base of the jet. E obs (keV) N o r m a li z ed P ho t on F l u x Configuration-IIIConfiguration-IVConfiguration-V E obs (keV) N o r m a li z ed P ho t on F l u x Configuration-IIIConfiguration-IVConfiguration-V
FIG. 2. Iron line shapes of Kerr BHs with Proca hair in the configurations III (solid lines), IV (dashed lines), and V (dottedlines) assuming the intensity profile ∝ /r (left panel) and ∝ h/ ( h + r ) / with h = 2 and r and h in units of the Procafield mass µ (right panel). The viewing angle is i = 45 ◦ and the energy of the line in the rest frame of the emitting gas is at6.4 keV. The photon flux is normalized in such a way that if we integrate over the energy bins we obtain 1. As it is evident from Fig. 2, the shape of the iron linestrongly depends on the emissivity profile. In particular,if we assume the intensity profile 1 /r , the solutions IVand V have an iron line with a high photon count at lowenergies, which is not the case for the iron lines of KerrBHs in vacuum. This happens because in the case ofa Kerr BH with Proca hair the photons can be emittedcloser to the center (so they can be significantly gravi-tationally redshifted) and can still escape to infinity. Ifwe change the intensity profile, we cannot change themaximum and minimum energies of the photons reachingthe distant observer (only determined by the backgroundmetric), but we can change the shape of the line. In thisway we can decrease the photon count at low energies, asshown in the right panel in Fig. 2. As already stressed inprevious work, see in particular Ref. [36], accurate testsof the metric around astrophysical BHs will only be possi-ble in the presence of the correct astrophysical model forthe corona. Phenomenological metrics, like a power-lawor a broken power-law, are approximations unsuitable to get strong and reliable constraints on the spacetime met-ric with the high quality data expected with the nextgeneration of X-ray facilities. IV. SIMULATIONS
We simulate observations with XIS/Suzaku , as an ex-ample for studying the capabilities of current X-ray mis-sions, and LAD/eXTP [57], in order to explore the op-portunities offered by the next generation of X-ray satel-lites. We employ the strategy already used in our studiesof Kerr BHs with scalar hair [2], boson stars [35], andProca stars [36], as well as to the study of other met-rics [58, 59]. We simulate observations of our metrics and http://heasarc.gsfc.nasa.gov/docs/suzaku/ we then fit the data with a Kerr model. If we find a goodfit, this means that we cannot distinguish our model froma Kerr spacetime. If the fit is not acceptable, the modelcan potentially be tested with the mission in considera-tion. Current X-ray data of BHs are consistent with theKerr metric, so a non-acceptable fit from XIS/Suzakushould rule out the spacetime. A non-acceptable fit withLAD/eXTP can instead be interpreted as the fact thatsuch a mission can test that configuration. Note thatthis is a simple study, but we believe it can catch the keyresults and establish a proof of concept.We use Xspec with the redistribution matrix,ancillary, and background files of XIS/Suzaku andLAD/eXTP following the forward-folding approach com-mon in X-ray astronomy. The spectrum measured by aninstrument is given as a photon count per spectral binand can be written as C ( h ) = τ (cid:90) R ( h, E ) A ( E ) s ( E ) dE , (7)where h is the spectral channel, τ is the exposure time, E is the photon energy, R ( h, E ) is the redistribution matrix(essentially the response of the instrument), A ( E ) is theeffective area (say the efficiency of the instrument, whichis given in the ancillary file), and s ( E ) is the intrinsicspectrum of the source. In general, the redistributionmatrix cannot be inverted, and for this reason we haveto deal with C ( h ) (folded spectrum). With the forward-folding approach, we consider a set of parameter valuesfor the intrinsic spectrum, we find C ( h ), we compare the“observed spectrum” with the folded spectrum with somegoodness-of-fit statistical test (e.g. χ ), and we repeat allthese steps to find “the best fit” by changing the inputparameters in the theoretical model.Our simulations are done assuming the intrinsic spec-tra s ( E ) calculated in the Kerr spacetimes with Procahair in the previous section and we add the background.The latter includes the noises of the instrument andof the environment (e.g. photons not from the targetsource or cosmic rays). The observational error has alsothe intrinsic noise of the source (Poisson noise) due tothe fact the spectrum is as a photon count per bin andis not a continuous quantity. The main advantage ofLAD/eXTP with respect to current X-ray instruments isthe much larger effective area of the detector. The effec-tive area of LAD/eXTP at 6 keV is about 35,000 cm .The effective area of XIS/Suzaku at 6 keV is about250 cm . This means that, for the same source andthe same exposure time, the photon count at 6 keV ofLAD/eXTP is 35 , / ≈
150 times the photon countof XIS/Suzaku. This significantly reduces the Poissonnoise and leads a more precise measurement of the spec-trum. Xspec is a package for X-ray data analysis. More details can befound at https://heasarc.gsfc.nasa.gov/xanadu/xspec/
For these simulations, we do not consider a specificsource, but we employ typical parameters for a bright BHbinary, namely an X-ray binary with a stellar-mass BHand a stellar companion. This is likely the most suitablecandidate for this kind of tests, because they are brighterthan supermassive BHs and therefore we can get a moreprecise measurement of the spectrum. We assume thatthe energy flux in the 1-10 keV range is 10 − erg/s/cm .The spectrum of the source is a power-law with photonindex Γ = 1 . ∝ E − Γ ) to describe theprimary component from the corona and a single ironline with an equivalent width of 200 eV. The iron linesused in these simulations are those shown in Fig. 2, sowe have six models, namely three different spacetimes(configurations III, IV, and V) and two types of intensityprofiles. The inclination angle of the disk is 45 ◦ . For bothmissions, we assume an exposure time of 100 ks.The simulations are then treated as real data. Afterrebinning to ensure a minimum count rate per bin of 20in order to use the χ statistics, we fit the data with apower-law and an iron line for a Kerr spacetime. For thelatter, we use RELLINE [60], which is an Xspec model todescribe a single relativistically broadened emission linefrom the accretion disk of a Kerr BH. We have 9 freeparameters: the photon index of the power law Γ, thenormalization of the power-law, inner and outer emis-sivity indices q and q , the breaking radius r br , the spinparameter a ∗ , the inclination angle of the disk i , the outerradius of the disk, and the normalization of the iron line.Fig. 3 shows our results for XIS/Suzaku, while Fig. 4is for LAD/eXTP. In each figure, the left panels are forthe power-law profile 1 /r and the right panels are forthe lamppost-inspired profile. The top panels shows theresults for Configuration III, the central panels those forConfiguration IV, and the bottom panels are for Config-uration V. In every panel, the top quadrant shows thefolded spectrum of the simulated data and the best-fit,while to bottom quadrant shows the ratio between thedata and the best fit. The larger relative errors on thephoton count at higher energies (particularly evident inFig. 3) is due to the lower photon count (Poisson noise)at high energies. In the case of XIS/Suzaku, the effectivearea of the instrument drops above 7 keV and thereforethe ratio between the model and the best fit becomesmuch noisier. Note that in Figs. 3 and 4 the y axis of thetop quadrant in each panel is in units of counts/s/keVand therefore it cannot be directly compared with thevalues of the y axis in Fig. 2, where the flux is normal-ized. The word “normalized” in the y axis in Figs. 3and 4 refers to properties of the response file that canbe ignored here (more details can be found in the Xspecmanual).The summary of the results with the best-fits is re-ported in Tab. II for XIS/Suzaku and in Tab. III forLAD/eXTP. The uncertainties on the parameters haveto be taken with some caution, because they are not veryrelevant in this study for our conclusions and thereforethey have not been estimated accurately. Configuration Profile χ , red a ∗ i q q r br r out III PL 1.06 0.91(1) 45(1) 7(1) 2.4(4) 4.3(5) 156(66)IV PL 1.04 > .
99 57(2) 8.4(4) – – –V PL 1.09 0.974(2) 21(1) 9.6(2) – – –III LP 1.04 0.96(15) 45.5(5) 2.1 – – –IV LP 0.98 0.96(1) 46.7(8) 3.7(1) – – –V LP 1.04 > .
99 46(1) 3.7(3) – – –TABLE II. Summary of the 100 ks simulations with XIS/Suzaku. In the second column “profile”, PL stands for power-law(profile ∝ /r ) and LP stands for lamppost [profile ∝ h/ ( h + r ) / with h = 2]. The third column is the value of the reduced χ of the best-fit. From the fourth to the nineth columns, we show the best-fit values of the spin parameter ( a ∗ ), the viewingangle ( i ), the inner emissivity index ( q ), the outer emissivity index ( q ), the breaking radius in gravitational radii ( r br ), andthe outer edge of the emission region in the accretion disk in gravitational radii ( r out ). When it is not possible to constrain theparameter, there is –. The uncertainties on the estimate of the parameters are the values in the round brackets and have to betaken only as a general guide. See the text for more details.Configuration Profile χ , red a ∗ i q q r br r out III PL 1.15 0.931(2) 44.83(6) 3.98(9) 3.28(5) 4.2(3) 104(23)IV PL 31 > .
99 59.1(3) 7.82(6) 4 – 3.24(7)V PL 257 > .
99 31.5(2) 10 3.98(3) 3.02(2) 20.7(6)III LP 3.43 0.923(4) 45.39(4) 10 2.12(2) 2.7(1) 58.4(9)IV LP 3.01 0.895(2) 45.59(9) 3.78(2) 3.7(4) – –V LP 4.02 0.989(4) 45.67(8) 8 3.67(3) – 27(4)TABLE III. As in Tab. II for the simulations with LAD/eXTP. See the text for more details.
V. DISCUSSION
The interpretation of the simulations with XIS/Suzakuare quite straightforward. The reduced χ of the best-fitis close to 1 and the Kerr model can well fit the data. Theratio between the simulated data and the best-fit in thebottom quadrant of each panel in Fig. 3 is always close to1 and we do not see unresolved features. The conclusionis that current X-ray missions cannot distinguish config-urations III, IV, and V from a Kerr BH. Note that wehave considered a quite bright source (flux in the energyrange 1-10 keV of 10 − erg/s/cm ) and a relatively longexposure time (100 ks). Moreover, the spectrum is a sim-ple power-law with an iron line. These should be quitefavorable ingredients for constraining the metric, and thefact that the Kerr model is capable of providing a goodfit does not leave much space to other conclusions.In the case of the 1 /r intensity profile, the iron lineshape are those shown in the left panel of Fig. 2. Those ofConfiguration IV and, in particular, of Configuration Vare definitively different from those expected from theline computed in the Kerr metric with the 1 /r intensityprofile (see, for instance, Fig. 9 in Ref. [33]). Neverthe-less, it is possible to find a good fit with a Kerr modelemploying a large value for the inner emissivity index.The latter indeed has the effect to increase the relativecontribution from the radiation emitted from the innerpart of the disk, which is strongly redshifted, and there- fore it has the capability of reproducing the peak in theleft panel of Fig. 2 for the iron shape of Configuration V(see again Fig. 9 in Ref. [33]). On the contrary, if we“knew” that the actual intensity profile goes like 1 /r ,the fit would be horrible. In other words, there is adegeneracy between the intensity profile and the back-ground metric, which could be fixed only in the case ofa good understanding of the former (which, in turn, isdetermined by the geometry of the corona).Let us now move to the simulations with LAD/eXTP.When we assume the 1 /r intensity profile, the spectraof Configuration IV and Configuration V cannot be fit-ted with a Kerr model. The minimum of the reduced χ is a large number and there are no ambiguities. Inthe case of Configuration III, the fit is good and the esti-mate of the parameters is not far from their actual value.In other words, Configuration III is too similar to thespacetime of a Kerr BH and the impact of its Proca hairis too weak. If we freeze the emissivity profile to its ex-act form, we find that the Kerr model cannot provide agood fit, see Fig. 5. The reduced χ of the best-fit isabout 4.2 and we see unresolved features in the ratio be-tween the simulated data and the best-fit. Once again,this shows that accurate tests of the Kerr metric usingthe iron line method will only be possible when the ge-ometry of the corona is known and we have a theoreticalprediction of the intensity profile of the reflection spec-trum. Phenomenological models, like a power-law or a no r m a li z ed c oun t s s − k e V − data and folded model r a t i o Energy (keV) no r m a li z ed c oun t s s − k e V − data and folded model r a t i o Energy (keV) no r m a li z ed c oun t s s − k e V − data and folded model r a t i o Energy (keV) no r m a li z ed c oun t s s − k e V − data and folded model r a t i o Energy (keV) no r m a li z ed c oun t s s − k e V − data and folded model r a t i o Energy (keV) no r m a li z ed c oun t s s − k e V − data and folded model r a t i o Energy (keV)
FIG. 3. Best-fits for Configuration III (top panels), Configuration IV (central panels), and Configuration V (bottom panels)from the simulations with XIS/Suzaku. The intensity profile of the reflection component in these simulations is described bythe power-law 1 /r (left panels) and the lamppost-inspired model h/ ( h + r ) / with h = 2 (right panels). In every panel, thetop quadrant shows the simulated data and the Kerr-model best-fit (folded spectrum), while the bottom quadrant shows theratio between the simulated data and the Kerr-model best-fit. See the text for more details. broken power-law often used with current data, are notadequate in the presence of high quality data possiblewith the next generation of X-ray facilities.Last, we have the simulations with LAD/eXTP withthe lamppost-inspired intensity profile. In this case, theiron lines can better mimic those expected in a Kerr met-ric assuming an intensity profile described by a brokenpower-law. Again, this can be easily interpreted withthe fact that stringent tests of the metric around BHswill only be possible if we can predict theoretically theintensity profile from the geometry of the corona. In allthe three spacetimes, the best-fit is bad ( χ , red > VI. CONCLUDING REMARKS
In this paper, we have extended previous work to ex-plore the capabilities of present and future X-ray mis-sions to test astrophysical BH candidates using X-rayreflection spectroscopy, the so-called iron line method,and in particular the possibility of distinguishing KerrBHs from Kerr BHs with Proca hair. The latter arenon-vacuum solutions of 4-dimensional Einsteins grav-ity, in which matter is described by a Proca (massivespin-1) field. The solutions can evade the no-hair theo-rem because of a time-dependence in the field, even if itsenergy-momentum tensor and the background metric arestationary.We have simulated observations with XIS/Suzaku, tostudy the capabilities of present X-ray missions, andLAD/eXTP, to understand the opportunities with thenext generations of X-ray facilities. We have simulatedobservations of a small set of Kerr BHs with Proca hairassuming the parameters for a bright BH binary, becausethe latter is likely the kind of source suitable for thesetests. The technique can also be applied to supermassiveBHs, but the constraints would be weaker because theirluminosity is lower and the Poisson noise is thus higher.The results of our simulations are shown in Fig. 3 (XIS/Suzaku) and Fig. 4 (LAD/eXTP) and summarizedin Tab. II (XIS/Suzaku) and Tab. III (LAD/eXTP). Inthe case of XIS/Suzaku, it is not possible to distinguishthese solutions from Kerr BHs: the fit is good, with χ , min close to 1 and no unresolved features. This istrue even for Configuration V with the 1 /r intensityprofile, because it may be interpreted as a Kerr BH witha high value of the inner emissivity index. We remarkthat our results do not exclude that very, or extremely,hairy BHs with Proca hair may exist whose reflectionspectrum cannot be mimicked by any intensity profile onthe Kerr background. To test this a more complete scan-ning of the parameter space must be performed. But itis quite instructive to realize how a BH which is con-siderably different from the standard Kerr solution canyield a similar phenomenology, as observed by currentinstruments. The situation is very different in the case ofLAD/eXTP, where usually the simulations cannot pro-vide good fits with a Kerr model. Configuration III withthe 1 /r intensity profile can still provide an acceptablefit.All these results have to be taken with caution, becauseof the numerous simplifications in our analysis and thelimited sample studied. However, as a proof of concept,they illustrate important features, such as the degener-acy between the metric and the intensity profile, even forextremely non-Kerr like metrics. As already stressed inRef. [36], we would get a significant improvement in thesetests if we could know the behavior of the intensity pro-file, because of this degeneracy. Moreover, the differencebetween current and future X-ray missions is clear. ACKNOWLEDGMENTS
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FIG. 5. As in the top left panel of Fig. 4, but with the emissivity index qq