Iron K α line of boson stars
Zheng Cao, Alejandro Cardenas-Avendano, Menglei Zhou, Cosimo Bambi, Carlos A.R. Herdeiro, Eugen Radu
IIron K α line of boson stars Zheng Cao, a Alejandro C´ardenas-Avenda˜no, b,c
Menglei Zhou, a Cosimo Bambi, a,d, Carlos A R Herdeiro, e Eugen Radu e a Center for Field Theory and Particle Physics and Department of Physics,Fudan University, 220 Handan Road, 200433 Shanghai, China b Programa de Matem´atica, Fundaci´on Universitaria Konrad Lorenz,Carrera 9 Bis No. 62-43, 110231 Bogot´a, Colombia c eXtreme Gravity Institute, Department of Physics,Montana State University, 59717 Bozeman MT, USA d Theoretical Astrophysics, Eberhard-Karls Universit¨at T¨ubingen,Auf der Morgenstelle 10, 72076 T¨ubingen, Germany e Departamento de F´ısica da Universidade de Aveiro andCenter for Research and Development in Mathematics and Applications (CIDMA),Campus de Santiago, 3810-183 Aveiro, PortugalE-mail: [email protected], [email protected],[email protected], [email protected], [email protected], [email protected]
Abstract.
The present paper is a sequel to our previous work [Y. Ni et al., JCAP 1607, 049(2016)] in which we studied the iron K α line expected in the reflection spectrum of Kerr blackholes with scalar hair. These metrics are solutions of Einstein’s gravity minimally coupledto a massive, complex scalar field. They form a continuous bridge between a subset of Kerrblack holes and a family of rotating boson stars depending on one extra parameter, thedimensionless scalar hair parameter q , ranging from 0 (Kerr black holes) to 1 (boson stars).Here we study the limiting case q = 1, corresponding to rotating boson stars. For comparison,spherical boson stars are also considered. We simulate observations with XIS/Suzaku. Usingthe fact that current observations are well fit by the Kerr solution and thus requiring thatacceptable alternative compact objects must be compatible with a Kerr fit, we find thatsome boson star solutions are relatively easy to rule out as potential candidates to explainastrophysical black holes, while other solutions, which are neither too dilute nor too compactare more elusive and we argue that they cannot be distinguished from Kerr black holes bythe analysis of the iron line with current X-ray facilities. Keywords: astrophysical black holes, boson stars, X-rays Corresponding author a r X i v : . [ g r- q c ] S e p ontents α line 44 Simulations and discussion 7 m = 0) 94.2 Rotating boson stars ( m = 1) 10 The current paradigm for the nature of astrophysical black hole (BH) candidates is that theyare well described by the Kerr metric [1]. Such a solution describes a rotating BH and iscompletely specified by only two parameters: the mass M and the spin angular momentum J of the compact object. On the theoretical side, this view is supported by the uniquenesstheorems (see [2] for a review), stating that, in 4-dimensional general relativity, the onlystationary, axisymmetric, asymptotically-flat, regular ( i.e. without geometric singularitiesor closed time-like curves on or outside the event horizon) solution of the vacuum Einsteinequations is the Kerr metric. Dynamically, any initial deviations from the Kerr geometryoccurring during gravitational collapse, should be radiated away through the emission ofgravitational waves [3]. Moreover, the only other traditionally accepted degree of freedom– electric charge – can be, most likely, ignored, because a neutral equilibrium should bereached fairly quickly due to the highly ionized host environments of these objects, renderingany residual electric charge too small (for macroscopic bodies) to appreciably affect the space-time metric [4–6]. Also, the possible presence of an accretion disk should have a negligiblegeometric backreaction, because the disk mass is typically many orders of magnitude smallerthan the mass of the BH candidate [7, 8]. In the end, deviations from the Kerr metric seemonly to be possible in the presence of new physics, thus making their detection potentiallyvery rewarding.Consequently, in the past 10 years, there has been a significant work to study howpresent and future facilities can test the Kerr BH hypothesis, both with electromagneticobservations [9–12] and gravitational waves [12–15]. Among the electromagnetic approaches,the analysis of the iron K α line in the X-ray reflection spectrum of the accretion disk isa particularly powerful and promising tool to probe the strong gravity region around BHcandidates [16–23]. The shape of the iron K α line is significantly affected by relativisticeffects (Doppler boosting, gravitational redshift, light bending) occurring in the vicinity ofthe compact object. In the presence of high quality data and with the correct astrophysicalmodel, this technique promises to provide stringent constraints on possible deviations fromthe Kerr metric because it can break the parameter degeneracy between the spin and possiblenon-Kerr features [24–26]. Other techniques meet, by contrast, serious difficulties to breaksuch a parameter degeneracy [27–30]. – 1 –ests of the Kerr BH hypothesis fall, at least, into two general categories. The first oneconsiders novel exact solutions of general relativity (or even modified gravity), by analysingmore general matter contents beyond (electro-)vacuum, thus capable of producing non-Kerrmetrics (see e.g. [31–37]). The second one designs parametrized families of metric deforma-tions from Kerr [38–45], without worrying about which model they solve. Within the firstapproach, Kerr BHs with scalar hair (KBHsSH) [34] have recently gained a lot of attentiondue to their physically reasonable and astrophysically plausible matter sources. These solu-tions are regular on and outside an event horizon, and also obey all energy conditions. Theyare exact (albeit numerical) solutions of Einstein’s gravity minimally coupled to a massive,complex scalar field, and interpolate between (a subset of) Kerr BHs – when a normalized“hair” parameter, q , vanishes – and a family of gravitating solitons [47] – when q is maximal( q = 1) –, the so-called boson stars; see, e.g., Ref. [48] for a review. They can bypass thevarious no-scalar-hair theorems that apply to this model by combining rotation with a har-monic time-dependence in the scalar field – see [49] for a review of such theorems. Althoughthese solutions are found numerically [34], a proof of their existence in a small neighborhoodof the Kerr family is available [50].In Ref. [51], we have computed the iron K α line expected in the reflection spectrumof a small sample of KBHsSH with q <
1, to check whether present and future X-ray mis-sions can constrain the scalar charge of astrophysical BH candidates. We found that someKBHsSH would have an iron line definitively different from those seen in the spectrum of BHcandidates, and therefore such solutions are at tension with current data. Other considerablyhairy BHs, however, are compatible with current data, but future X-ray mission will be ableto put much stronger constraints.This work is the continuation of the study presented in Ref. [51]. Here we considerthe limiting case with q = 1, in which there is no horizon and the solution describes aneverywhere smooth, topological trivial configuration, i.e. a boson star. As in Ref. [51], westudy the iron line profile that should be expected in the reflection spectrum of the accretiondisk around these objects. We simulate observations with XIS/Suzaku to check whether itis possible to test the existence of these objects with the current X-ray facilities. We studya sample of 12 boson star solutions, whose lensing – another property of phenomenologicalinterest – was considered in [52]. We find that some solutions can be surely ruled out, asKerr models cannot provide a good fit of their iron line, while they do it with current X-raydata. Other solutions, assuming certain emissivity profiles, can have an iron line too similarto that of Kerr BHs to be tested by current X-ray missions. Last, there are a few solutionsthat may be tested by current X-ray facilities, but we cannot conclude they can be ruled outby current data without a more sophisticated analysis. We remark that boson stars havebeen suggested as BH mimickers, and studies in this context, including of the comparativephenomenology, have been report in, e.g. [53–64].The content of the paper is as follows. In Section 2, we briefly review the boson starsolutions. We then describe the sample of 12 solutions that shall be studied in this paper. InSection 3, we compute the shape of the iron line profile expected in the reflection spectrumof these 12 spacetimes. In Section 4, we simulate observations with XIS/Suzaku and westudy whether these boson starts solutions can be tested with present X-ray observatories.Summary and conclusions are presented in Section 5. Throughout the paper, we employnatural units in which c = G N = (cid:126) = 1. http://heasarc.gsfc.nasa.gov/docs/suzaku/ – 2 – Boson stars
John Wheeler famously put forward the idea of “geons” [65], proposing that elementary par-ticles admitted a fundamental description as geometric-electromagnetic entities, made purelyof gravitational and electromagnetic fields. Unfortunately, within classical Einstein-Maxwelltheory, no everywhere regular, stationary, asymptotic flat solutions describing localized lumpsof energy exist [68], at least with trivial topology, rendering Wheeler’s vision unphysical (suchconfigurations, however, could be found for other asymptotics [66, 67]).Still, Wheeler’s proposal motivated much further work. In particular, Kaup decidedto search for “geons” in scalar-vacuum, rather than electrovacuum [69]. In order to avoidno-go Derrick-type theorems, Kaup included a time dependence for the scalar field, which,for it to be compatible with a static geometry, was chosen as a harmonic time-dependence,as for stationary states in quantum mechanics, and the scalar field was taken to be complex.Then, for such a complex scalar field, if a mass term is present, the corresponding Einstein-Klein-Gordon model admits geon-type solutions (also constructed in [70]), which in modernlanguage are called boson stars , and regarded as an example of a gravitating soliton. In anutshell, the harmonic time dependence provides an effective pressure that can balance thesystem’s self-gravity and prevent gravitational collapse into a BH, up to a maximal totalmass, that depends on the scalar field mass.Over the years, several generalizations of the original boson stars have been discussed,considering, for instance, self-interactions for the scalar field, and many of their physicalproperties have been analysed – see [46–48] for reviews. In particular, a subset of bosonstars are stable against perturbations and have been shown to form dynamically; rotating,stationary boson stars (rather than static, spherical ones) have also been constructed, startingwith [71, 72] (see also [36, 73–78]) and analogous solutions made up of massive vector (ratherthan massive scalar) fields have, furthermore, recently been shown to exist [79].Boson star solutions in the Einstein-(massive, complex)Klein-Gordon system are foundby taking, besides the appropriate (static or stationary) ansatz for the metric, a scalar fieldansatz of the form: Φ( t, r, θ, ϕ ) = φ ( r, θ ) e − iwt + imϕ , (2.1)where φ is the profile function, w is the scalar field harmonic frequency and m ∈ Z + is theazimuthal harmonic index; also, r, θ, ϕ are spherical-type coordinates, whereas t is the timecoordinate. Spherical boson stars have m = 0 and the scalar profile only depending on theradial coordinate, r . For m (cid:62) m = 1 case. Also, all solutions to be considered here are nodeless, meaning that thescalar field profile has no nodes.An informative way to see the domain of existence of these boson stars is an ADMmass M vs. scalar field frequency diagram, as that shown in Fig. 1. In this plot, the redsolid curve describes the rotating boson star solutions whereas the red dashed curve describesthe static ones. Overall, 12 particular solution points are highlighted corresponding to thespecific solutions to be analized below, whose basic properties are described in Tab. 1. Theseare the same solutions that have been analysed in the context of lensing, in [52]. The axesare shown in (Planck units and) units of the scalar field mass µ . Thus, the vertical axis maybe regarded as (half of) the Schwarzschild radius of the boson star over the Compton wavelength of the scalar field. One can see that it cannot be much larger than unit. In otherwords, when there is too much scalar mass, such that the Compton wave length of the scalar– 3 – µ M AD M w/ µ m=1 m=0 Figure 1 . Domain of existence of the boson star solutions in an ADM mass vs. scalar field frequencydiagram. The red solid (dashed) line describes the family of rotating boson stars with m = 1 (sphericalboson stars with m = 0). The twelve highlighted points correspond to the configurations to beanalysed below. The same spiraling pattern holds for higher m , cf. Fig. 1 (left) in [35]. field is much smaller than the corresponding Schwarzschild radii, collapse cannot be avoided,and there are no boson star solutions.When following the ( w, M ) spiral, one can distinguish different regions. First, there isa region of weak gravity, with w close to µ (solutions 1 and 6), where the solutions are quitespread out, with an effective size much larger than their Schwarzschild radii, and a Newtoniandescription is accurate. Moving further along either of the spirals, the compactness of theboson stars increases. After the maximal ADM mass is attained, the solutions are expectedto become unstable against perturbations (see e.g. [35]). Thus, solutions 3,4,5 and 9 to 12 areexpected do be unstable, even though, in the rotating case, the time scales are not preciselyknown. Also, when the solutions are sufficiently compact, they develop light rings. For therotating case this happens for solutions 10-12. In the spherical case, this only occurs beyondsolution 5 and hence none of the spherical solutions possess light rings. We also remark thatthe solutions can be uniquely labeled by the maximal value, φ max , of the scalar profile φ ,which increases monotonically along the boson star curve (with φ max → w → µ ) [35]. α line Broad iron K α lines are a common feature in the X-ray spectrum of astrophysical BH candi-dates [80–83]. In its rest-frame, this is a very narrow line around 6.4 keV for neutral iron, andshifts up to 6.97 keV for H-like iron ions. The line in the X-ray spectrum of BH candidatesis instead broad and skewed due to special and general relativistic effects (Doppler boosting,gravitational redshift, light bending) occurring in the strong gravity region near the compactobject. It is thus thought that, if properly understood, the iron K α line can be a powerfulprobe to test the metric of the strong gravity region [24–26].– 4 –olution m w M J r ISCO r r r Table 1 . Properties of the boson star solutions 1-12. The sixth columns (in units of µ ) report the valueof the ISCO radius: the non-rotating solutions 1-5 have no ISCO (circular orbits are always stable),while the rotating solutions 6-12 have an ISCO, which is the same for corotating and counterrotatingorbits. However, in the case of counterrotating orbits, the solutions 7-9 have stable orbits for r between r ISCO and r , as well as for r > r , while the orbits are unstable for r between r and r . In thecase of the solutions 10-12, counterrotating orbits are stable for r between r ISCO and r , there are nocircular orbits (stable or unstable) for r between r and r , there are unstable circular orbits for r between r and r , and circular orbits are again stable for r > r . Within the disk-corona model [84, 85], the compact object is surrounded by a thinaccretion disk, which is described by the Novikov-Thorne model [86]. The disk is in theequatorial plane, perpendicular to the spin of the central body. The particles of the gasfollow nearly geodesic circular orbits. The inner edge of the disk is at the ISCO radius r ISCO .When the particles of the gas reach the ISCO, they quickly plunge onto the central objectwithout emitting additional radiation. The disk emits as a black body locally, and as amulti-color black body when integrated radially. The temperature of the inner edge of thedisk is ∼ ∼
10 eV for supermassiveBH candidates. The “corona” is a hotter ( ∼
100 keV), usually optically-thin, electron cloud,which enshrouds the disk. Due to inverse Compton scattering of the thermal photons fromthe disk off the hot electrons in the corona, the latter becomes an X-ray source with a power-law spectrum. Some X-ray photons of the corona illuminate the disk and produce a reflectioncomponent with some emission lines. The iron K α line is usually the most prominent featurein the reflection spectrum of the disk.In the Novikov-Thorne model, the particles of the gas follow nearly geodesic equatorialcircular orbits, and therefore the angular velocity of the disk at any radius is given bythe corresponding Keplerian angular velocity Ω K . In the Kerr spacetime and in many otherspacetimes, d Ω K /dr <
0, and the magneto-rotational instability (MRI), which is the standardmechanism to drive the accretion process in Keplerian disks, can work. However, there areexamples of non-Kerr spacetimes where there are at least regions with stable equatorialcircular orbits where d Ω K /dr < d Ω K /dr ,because we have the metric in numerical form. For corotating disks, the condition d Ω K /dr < r > r ISCO usually holds. Among the 12 solutions discussed here, only for the solution 6the condition is not always satisfied for radii r > r
ISCO (but it is satisfied for sufficiently largeradii). For the solution 6, the existence of a Keplerian thin disk with the inner edge at theISCO radius requires the presence of some other mechanism to drive the accretion processat very small radii. In what follows, when we consider the configuration 6, we will assumethat such a mechanism exists.The shape of the iron K α line in the X-ray reflection spectrum of the disk is determinedby the metric around the compact object, the inclination angle of the disk with respect tothe line of sight of the observer i , the geometry of the emission region, and the emissivityprofile of the disk. The emission region is the accretion disk, from its inner edge r in to someouter edge r out .The emissivity profile plays an important role in the final iron line shape and could becalculated theoretically if we knew the exact geometry of the corona. However, this is not thecase at the moment, and therefore it is common to employ phenomenological descriptions.The simplest possibility is to model the intensity profile with a power law, i.e. to assumethat the specific intensity of the radiation in the rest-frame of the gas in the accretion diskis I e ∝ /r α , where r is the radius of the emission point and α is the emission index. Thelatter can be a free parameter to be determined by the fit. A more sophisticated model is abroken power-law, i.e. I e ∝ (cid:26) /r α , for r < r b , /r α , for r > r b . (3.1)Here we may have three free parameters: two emissivity indexes, α and α , and the brokenradius r b . It is also common to assume two free parameters, α and r b , and set α = 3,which corresponds to the Newtonian limit at large radii in the lamppost geometry.We calculate the iron lines expected in the reflection spectrum of the accretion disksof the boson star solutions 1-12 with the code described in Refs. [19, 90] and extended inRef. [91] for numerical metrics . The results are shown in Fig. 2. The viewing angle is i = 20 ◦ (top panels), 45 ◦ (central panels), and 70 ◦ (bottom panels). The local intensity ofthe radiation is I e ∝ /r . For the non-rotating solutions 1-5 without ISCO, we have assumed r in = 0. For the rotating solutions 6-12, we have r in = r ISCO . Here and in what follows, wealways assume that the disk is correlating with the spin of the central object. In all thesesimulations, we have r out = 25 in units in which 1 /µ = 1. In order to have r out in terms ofthe gravitational radius of the object ( r g = M ), it is necessary to divide r out by the ADMmass in the fourth column of Tab. 1. Such a low value of r out will be employed even later inthe simulations with XIS/Suzaku because it is computationally quite expensive to computethe iron line profile for a larger outer radius. However, we stress that our results do notdepend on the choice of r out because the statistics in our simulations is not good enough topermit the measurement of the outer edge of the accretion disk.As the power-law emissivity profile 1 /r is not very physical if r in = 0 or very small, wehave calculated also the iron line profiles of the solutions 1-5 assuming the lamppost intensity Our calculations do not include possible ghost images of the accretion disk. If the central object is a BH,the effect, if any, is very small. If the central object is not a BH, the effect might be more important [92].In the semi-quantitative analysis in the next section, we can only distinguish iron lines significantly differentfrom those expected in the Kerr metric, and therefore we do not expect that the contribution from ghostimages can alter our results and conclusions. – 6 –rofile in flat spacetime. In this case we have [93] I e ∝ h ( h + r ) / . (3.2)Fig. 3 shows the iron line profile for the solutions 2 and 4, varying also the inner edge ofthe accretion disk r in (we consider a disk truncated as in Refs. [21, 94]). The solutions 1and 3 are quite similar to the solution 2, while the solution 5 is similar to the solution 4. Inthe solutions 1-3, the photons emitted at very small radii are not strongly redshifted, so theimpact on the iron line profile of a finite r in is moderate. In the solutions 4 and 5, photonsemitted at very small radii are strongly redshifted: if we change the emissivity profile from ∝ /r to Eq. (3.2), we remove the peak at low energies and the shape of the iron line looksmuch more like that expected in the Kerr metric. In this section we want to understand whether the analysis of the iron line profile can dis-tinguish boson stars and Kerr BHs, and in particular if the available X-ray data can alreadyrule out some boson star solutions. To do this, we proceed as in Ref. [51]. Our starting pointis that the available reflection models on the market assume the Kerr metric and can providegood fits of the reflection spectrum of BH candidates observed with current X-ray facilities.We thus perform simulations of the X-ray spectrum expected in the case of the boson starsolutions 1-12. We fit the simulated observations with a Kerr model. If the latter provides agood fit, we conclude that a similar observation cannot distinguish a boson star from a KerrBH. If the fit is bad, we argue that astrophysical BHs cannot be such a boson star, becausethere is no tension between current observational data and theoretical reflection models inthe Kerr metric. This is a simple analysis, but we expect that it is able to provide the rightinsight. Of course, if any future X-ray observation yields data incompatible with the Kerrmodel, our conclusions must be revisited.We do not consider a specific source, but we assume some typical parameters for abright BH binary, as a binary is a more suitable source than an AGN for this kind of tests.The X-ray spectrum of the BH binary is approximated by a single iron K α line and a power-law component. In principle, we should consider the whole reflection spectrum rather thana single iron line, but this is enough for a preliminary analysis, as the information on thespacetime geometry is mainly encoded in the iron line. In our simulations, the flux in the0.7-10 keV range is about 4 · − erg/s/cm and the iron line equivalent width is about200 eV. The photon index of the power-law component is Γ = 2. These are the same valuesas those employed in Ref. [51].We use XSPEC with the background, the ancillary, and the response matrix files ofXIS/Suzaku to simulate the data. For every boson star solution, we simulate four sets ofobservations: the time exposure can be either 100 ks or 1 Ms, while the emissivity of theiron line is calculated either assuming the power-law profile ∝ /r or the lamppost profilein Eq. (3.2). The photon count in the 0.7-10 keV range turns out to be about 3 · and3 · , respectively for the 100 ks and 1 Ms observations. For every set of observations, weconsider three possible viewing angles, i = 20 ◦ , 45 ◦ , and 70 ◦ . However, we will see that theviewing angle cannot qualitatively change the fit and thus plays only a marginal rule in ourdiscussion. http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/index.html – 7 – P ho t on F l u x E obs (keV)i = 20 o P ho t on F l u x E obs (keV)i = 20 o P ho t on F l u x E obs (keV)i = 45 o P ho t on F l u x E obs (keV)i = 45 o P ho t on F l u x E obs (keV)i = 70 o P ho t on F l u x E obs (keV)i = 70 o Figure 2 . Broad iron K α lines expected in the X-ray reflection spectrum of the accretion disksaround the boson star solutions 1-5 (left panels) and 6-12 (right panels). The inclination angle ofthe disk with respect to the line of sight of the observer is i = 20 ◦ (top panels), 45 ◦ (central panels),and 70 ◦ (bottom panels). The local spectrum has the intensity profile 1 /r . The numbers in the toppanels indicate the boson star solution. For every solution, we use the same line style in the centraland bottom panels. Photon flux in arbitrary units. The simulated data are fitted using XSPEC with a power-law plus a single iron line.The latter is introduced with KERRCONV*GAUSSIAN, which assumes the Kerr metric andthe broken power-law profile in Eq. (3.1). For the power-law, we have two free parameters:the photon index of the power-law Γ and the normalization of the power-law. For the iron– 8 – P ho t on F l u x E obs (keV)Solution 2, i = 45 o r in = 0r in = 0.5r in = 1r in = 2r in = 4 P ho t on F l u x E obs (keV)Solution 4, i = 45 o r in = 0r in = 0.5r in = 1r in = 2r in = 4 Figure 3 . Broad iron K α lines expected in the X-ray reflection spectrum of the accretion disksaround the boson star solutions 2 (left panel) and 4 (right panel), assuming the lamppost intensityprofile h/ ( h + r ) / and different values of the inner radius r in . The inclination angle of the diskwith respect to the line of sight of the observer is i = 45 ◦ . Photon flux in arbitrary units. line, we have six free parameters: the BH spin a ∗ , the inclination angle of the disk withrespect to the line of sight of the observer i , the emissivity index α , the emissivity index α ,the breaking radius r b , and the normalization of the iron line.In Ref. [51] we also simulated observations with LAD, which is expected on board of thefuture X-ray mission eXTP [95]. Here we do not consider LAD/eXTP, because its statisticsis so good that we would need the correct (or at least a plausible) emissivity profile for theiron line. However, in the case of the boson stars without ISCO we do not have a goodmodel for the intensity profile and we thus prefer to omit the discussion of possible testswith LAD/eXTP. We also note that the outer radius of the accretion disk in our simulationsis set to 25 (see previous section), while we fit the simulated data with an iron line modelassuming a fixed outer radius r out = 400 M . We have checked that such a simplificationdoes not appreciably change our fits for the simulated observations of 100 ks and 1 Ms withXIS/Suzaku. On the contrary, the exact value of the outer radius is important with the highquality data of LAD/eXTP. The effective area at 6 keV of XIS/Suzaku is about 800 cm , tobe compared with the ∼ ,
000 cm of LAD/eXTP, and this makes the difference. ( m = 0)We start simulating observations for the non-rotating boson star solutions 1-5. In the firstset of simulations, the exposure time is 100 ks and the emissivity profile is modeled by thepower-law 1 /r . These solutions have no ISCO, namely circular orbits are always stable. Wefirst consider the case in which there is no inner edge of the disk, in the sense that we set r in = 0 in the code. We find that the Kerr model always provide a very bad fit and it seemsthat the fit gets worse moving along the q = 1 curves from the solution 1 to the solution 5.For a given solution, we find that the minimum of the reduced χ is lower for i = 70 ◦ andhigher for i = 20 ◦ . Figs. 4 and 5 show the case of the solution 2, respectively for i = 20 ◦ and i = 70 ◦ . In each figure, the top panel is the folded spectrum, the bottom panel is theratio between the folded spectrum and the simulated data. For both viewing angles, there isa clear unresolved feature around 5 keV, suggesting that we are using the wrong model to fitthe data. The fits of the solutions 1-3 are bad, with the minimum of the reduced χ ranging– 9 –rom 2 to 10. The fits of the solutions 4 and 5 are even worse, and we find that the minimumof the reduced χ is more than 40.If we set the inner edge at a finite radius, like r in = 0 . I e ∝ /r and r in = 0, the iron line profile is essentially determined by the emission near the center r = 0. The iron line profiles of the solutions 1-3 are not strongly redshifted, despite mostof the emission comes from the region near the center. In this case, if we set the inner edgeof the disk at a finite radius, the iron line profile changes, but the effect is not so dramatic.The iron line profiles of the solutions 4 and 5 are very redshifted and we have a peak at lowenergies, because the profile 1 /r diverges at r = 0. Setting r in at a finite radius, we removethe peak, and the fit gets better.We then consider simulations with longer exposure time, 1 Ms. In this case, the Kerrmodel can never provide an acceptable fits for the solutions 1-5, no matter we change theinner edge of the disk at some finite radius.We move to the simulations in which we adopt the lamppost emissivity profile with I e ∝ h/ ( h + r ) / . If the exposure time is 100 ks, the fits of the solutions 1-3 are stillunacceptable with clear unresolved features, independently of the location of the inner edgeof the disk. Fig. 6 illustrates the case of the solution 2 as an example. The fits of the solutions4 and 5 are acceptable and without unresolved features with r in = 0.If we consider 1 Ms observations, the Kerr model can still provides a good fit for thespectra of the solutions 4 and 5. Fig. 7 shows the results for the solutions 4 assuming r in = 0.The minimum of the reduced χ is around 1.0 and the fit is indeed good. We do not see anyunresolved feature in the plot of the ratio between the folded spectrum and the simulateddata. The iron lines in the right panel in Fig. 3 looked indeed quite similar to the Kerr ones. ( m = 1)We repeat the analysis of the previous subsection for the seven rotating boson stars, thesolutions 6-12. The inner edge of the accretion disk is set at the ISCO radius, which isalways finite in this case, see Tab. 1. As before, we consider the three viewing angles i = 20 ◦ ,45 ◦ , and 70 ◦ .We start assuming the power-law emissivity profile 1 /r and simulate observations of100 ks. We fit the simulated spectra with the Kerr model. The fit of the solution 8 is good:the minimum of the reduced χ is close to 1 and there are no unresolved features in the ratiobetween the folded Kerr spectrum and the simulated data. As we move along the q = 1curve, the fit gets worse. The fits of the solutions 7, 9, and 10 are still acceptable. The fitsof the solutions 6, 11, and 12 are not. When we simulate observations of 1 Ms, we find thatthe Kerr model can never provide an acceptable fit to the simulated data. Fig. 8 shows thefolded spectrum and the ratio between the folded spectrum and the simulated data of thesolution 8. The fit is clearly bad and indeed the minimum of the reduced χ is around 2.5.The fits of the other solutions are worse.We repeat the simulations for the lamppost intensity profile, in which I e ∝ h/ ( h + r ) / .We find the same qualitative results as the simulations with the power-law intensity profile.If the exposure time is 100 ks, the Kerr model provides a good fit for the spectrum of thesolution 8, still an acceptable fit for the solutions 7, 9, and 10, while the fits is not acceptablein the case of the solutions 6, 11, and 12. Once again, when the exposure time of the– 10 – igure 4 . Folded spectrum and ratio between the folded spectrum and the simulated data for theboson star solution 2, assuming the viewing angle i = 20 ◦ and the emissivity profile 1 /r . Theexposure time is 100 ks and the minimum of the reduced χ is about 6. See the text for more details. Figure 5 . As in Fig. 4, but for the viewing angle i = 70 ◦ . The minimum of the reduced χ is about2 and we still observe unresolved features. See the text for more details. simulations is 1 Ms, the fit with the Kerr model always shows some unresolved features anda large minimum of the reduced χ . Fig. 9 shows the fit of the spectrum of the solution 8.As before, the other fits are worse. – 11 – igure 6 . Folded spectrum and ratio between the folded spectrum and the simulated data forthe boson star solution 2, assuming the viewing angle i = 45 ◦ and the lamppost intensity profile h/ ( h + r ) / . The exposure time is 100 ks and the minimum of the reduced χ is about 1.5. Seethe text for more details. Figure 7 . Folded spectrum and ratio between the folded spectrum and the simulated data forthe boson star solution 4, assuming the viewing angle i = 45 ◦ and the lamppost intensity profile h/ ( h + r ) / . The exposure time is 1 Ms and the minimum of the reduced χ is about 1.0. See thetext for more details. – 12 – igure 8 . Folded spectrum and ratio between the folded spectrum and the simulated data for theboson star solution 8, assuming the viewing angle i = 70 ◦ and the intensity profile 1 /r . The exposuretime is 1 Ms and the minimum of the reduced χ is about 2.5. See the text for more details. Figure 9 . Folded spectrum and ratio between the folded spectrum and the simulated data forthe boson star solution 8, assuming the viewing angle i = 45 ◦ and the lamppost intensity profile h/ ( h + r ) / . The exposure time is 1 Ms and the minimum of the reduced χ is about 2.3. See thetext for more details. – 13 – Concluding remarks
In the present paper we have continued our study to observationally test the existence in theUniverse of Kerr BHs with scalar hair, which are solutions of Einstein’s gravity minimallycoupled to a massive, complex scalar field found in Ref. [34]. We have considered the limitingcase q = 1, which yields rotating ( m = 1) boson stars. For comparison, here we have alsoconsidered non-rotating (spherically symmetric) boson stars ( m = 0).We have computed the profile of the iron K α line that should be expected from thereflection spectrum of the accretion disk of these objects. We have simulated observationswith XIS/Suzaku, assuming that the source is a bright BH binary, and we have fitted thespectrum with a Kerr model. Since the available X-ray data of the reflection spectrum ofBH binaries are regularly fitted with Kerr models, and there is no tension between data andtheoretical predictions, we argue that the BH candidates in the Universe cannot be the bosonstar solutions with a bad fit, while those with an acceptable fit cannot be distinguished byKerr BHs and are thus viable candidates. We remark, again, that should any future obser-vations of BH candidates yield spectra incompatible with the Kerr model, the conclusionsherein should be revisited.Solution 1 /r , t = 100 ks 1 /r , t = 1 Ms LP, t = 100 ks LP, t = 1 Ms1 × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × Table 2 . Summary of our results. × means that the Kerr model fails to give an acceptable fit. In thesecond and the third columns, the model assumes I e ∝ /r . In the fourth and the fifth columns, wehave I e ∝ h/ ( h + r ) / . The exposure time is 100 ks (second and fourth columns) and 1 Ms (thirdand fifth columns). See the text for more details. Our results are summarized in Tab. 2, where × is to indicate that the Kerr model fails togive an acceptable fit for the corresponding boson star solution, exposure time, and emissivityprofile. The solutions 1-3, 6, 11, 12 can be ruled out. Even for an exposure time of 100 ks, wefind that the Kerr model always provides a very bad fit. The iron line profiles of these bosonstars are not compatible with those in the available X-ray data of BH binaries. For sphericalboson stars, the qualitative conclusion is that acceptable solutions must be fairly compact.This qualitative behaviour also holds for the rotating solutions; in this case, however, boththe most dilute and the most compact cases are ruled out for these “short” exposure times.In a sense, our sample of solutions included more compact rotating than spherical bosonstars, as the latter do not possess a light ring whereas the former do (beyond solution 10).– 14 –erhaps too compact spherical boson stars can also be ruled out. Unfortunately, in this caseon has to go well into the spiral in order to get a light ring, which only appears in the thirdbranch of solutions, i.e. after the second backbending in frequency, wherein the accuracy ofsolutions becomes poorer.The Kerr model provides an acceptable fit for the solutions 7-10 and for an exposuretime of 100 ks, but fails to give an acceptable fit in the case of a longer observation of 1 Ms.In such a case, we should investigate these solutions better and analyze real data of specificsources to conclude whether the solutions 7-10 can be excluded or are consistent with theavailable data. Unfortunately, this is beyond the possibilities of the current version of ourcode.Last, we have the solutions 4 and 5. If we assume the lamppost emissivity profile ∼ h/ ( h + r ) / , even an observation of 1 Ms cannot distinguish these solutions from a KerrBH, in the sense that the Kerr model provides a good fit. We thus argue that it impossibleto test the existence of these objects with the current X-ray missions. Acknowledgments
ZC, MZ, and CB were supported by the NSFC (grants 11305038 and U1531117) and theThousand Young Talents Program. CB also acknowledges support from the Alexander vonHumboldt Foundation. AC-A acknowledges funding from the Fundaci´on Universitaria Kon-rad Lorenz (Project 5INV1161) and the NSF CAREER Grant PHY-1250636. CH and ERacknowledge funding from the FCT-IF programme. This work was partially supported bythe H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904, and by the CIDMA projectUID/MAT/04106/2013.
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