Iron Kα line of Proca stars
Tianling Shen, Menglei Zhou, Cosimo Bambi, Carlos A.R. Herdeiro, Eugen Radu
IIron K α line of Proca stars Tianling Shen, 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: January 2017)X-ray reflection spectroscopy can be a powerful tool to test the nature of astrophysical blackholes. Extending previous work on Kerr black holes with scalar hair [1] and on boson stars [2], herewe study whether astrophysical black hole candidates may be horizonless, self-gravitating, vectorBose-Einstein condensates, known as
Proca stars [3]. We find that observations with current X-raymissions can only provide weak constraints and rule out solely Proca stars with low compactness.There are two reasons. First, at the moment we do not know the geometry of the corona, andtherefore the uncertainty in the emissivity profile limits the ability to constrain the backgroundmetric. Second, the photon number count is low even in the case of a bright black hole binary, andwe cannot have a precise measurement of the spectrum.
I. INTRODUCTION
Today we have robust observational evidence for theexistence of dark and compact objects that can be nat-urally interpreted as black holes (BHs) – see e.g. [4].These BH candidates could, however, be something else,but only in the presence of new physics [5]. Stellar-massBH candidates have a mass
M > M (cid:12) , and are there-fore too heavy to be compact relativistic stars, at leastif composed by standard matter [6]. Supermassive BHcandidates of 10 -10 M (cid:12) are at the center of galax-ies and are too massive, compact, and old to be clustersof non-luminous bodies [7]. Moroever, the gravitationalwaves observed by LIGO are consistent with the signalexpected from the coalescence of two BHs in general rel-ativity [8], even though we have not yet entered the eraof precision gravitational wave spectroscopy.Within the framework of standard physics, the space-time metric around astrophysical BHs should be well ap-proximated by the Kerr solution. Nevertheless, macro-scopic deviations are predicted in a number of scenariosinvolving new physics, which typically can be divided intotwo classes: ( i ) modified gravity; or ( ii ) exotic matter(within general relativity). In the latter context, one ofthe most natural and interesting models for BH mimick-ers is that of a horizonless, self-gravitating Bose-Einsteincondensate of ultra-light bosons. The first such modelwas found long ago by Kaup [9] and Ruffini and Bonaz-zola [10], corresponding to scalar boson stars . The modelrequires only a minimal number of (physically reason-able) new ingredients: a massive complex scalar field (noself-interactions are required, even though they are possi-ble [11, 12]) minimally coupled to Einstein’s gravity; mo-roever, an open set of the domain of existence of theseboson stars are known to be stable and even form dy-namically (see [13, 14] for reviews). More recently, spin ∗ Corresponding author: [email protected]
Proca stars ,have also been found in Einstein’s gravity minimally cou-pled to a massive complex vector field [3]. As for theirscalar cousins, these solution can be either static or ro-tating and seem to mimic some, but not all, propertiesof the scalar case [15].In this letter, we consider the Proca star solutionsfound in Ref. [3]. In particular, we want to address thequestions whether Proca stars can be a viable alternativeto BHs to explain the observed dark and compact objectsin the Universe or, on the contrary, can be already ruledout by current astrophysical observations. We answerthis question by considering the shape of the iron K α line commonly observed in the reflection spectra of as-trophysical BH candidates. We find that non-compactconfigurations are not consistent with the available X-ray data, but the constraints are weak and more compactProca stars can well mimic Kerr BHs. II. PROCA STARS
Vector boson stars (a.k.a. Proca stars) are solutionsto Einstein’s gravity with a minimally coupled complexProca field of mass µ [3]. The action of the model reads S = S EH + (cid:90) d x √− g L Proca , (1)where S EH is the Einstein-Hilbert action and L Proca = − F αβ ¯ F αβ − µ A α ¯ A α , (2)where A is the (complex) potential 1-form, F = dA isthe field strength, and the overbar denotes the complexconjugate quantities.With appropriate anstaz [3], both static, sphericallysymmetric and stationary, rotating solution can be found,corresponding to macroscopic, self-gravitating lumps of a r X i v : . [ g r- q c ] J a n FIG. 1. Proca star solutions in an ADM mass vs. vectorfield frequency diagram. The red dashed line describes thefamily of spherical Proca stars with m = 0, while the redsolid line is for the family of rotating Proca stars with m = 1.The 12 highlighted points correspond to the configurationsstudied in this work. the Proca field. These are prevented from gravitation-ally collapsing by an effective pressure associated toa harmonic time dependence in the potential 1-form, A ∼ e − iwt + imϕ , where t and φ are the coordinates as-sociated to the timelike (at infinity) Killing vector field ∂/∂t and azimuthal Killing vector field ∂/∂φ . w ∈ R + isthe frequency of harmonic oscillation and m ∈ Z / { } isthe azimuthal quantum number. Observe that due to thecomplex nature of the field, both the t and φ dependenceof the Proca potential ansatz vanish at the level of theenergy-momentum tensor, making this ansatz compati-ble with a stationary and axi-symmetric geometry [16].In Fig. 1 we exhbit the existence domain for spher-ically symmetric ( m = 0) and rotating (with m = 1)Proca stars, in an ADM mass vs. Proca field frequencydiagram, both quantities being made dimensionless byusing the Proca field mass µ . Both types of solutions fallalong a spiral-type line in this diagram, similarly to whathappens to scalar boson stars and also for solutions withhigher m [3]. As a rule of thumb, the compactness of thesolutions typically increases as we move along this spiral,starting from the vacuum case (maximal value of w , min-imal value of M ) – see Fig. 2 in [16] for a more detailedanalysis of compactness, for the case of scalar boson stars.Also, the stability of the solutions depends on their lo-cation along the spiral. For the spherically symmetriccase it was shown in [3] that the set of solutions startingat the vacuum point and up to the maximal mass arestable; at the maximal ADM mass an unstable mode de-velps and solutions beyond this point are expected to beperturbatively unstable. For the rotating case, albeit nodetailed perturbative computation has been made, somegeneric arguments suggest a similar picture should hold(see Sec. 6.2 in [16] for such a discussion in the case ofscalar boson stars).In Fig. 1 we have highlighted 12 representative solu- Solution w branch M J tions, 5 for the spherically symmetric case and 7 for therotating case, corresponding to the sample of Proca starsthat will be studied in detail in this paper. Some prop-erties of these solutions are detailed in Table I. As a rel-evant property for the following we observe that time-like, co-rotating, circular, equatorial geodesics exist allthe way up to the origin for all solutions studied. In otherwords there is no innermost (stable or unstable) circulargeodesic, for co-rotating orbits (the counter-rotating caseis more complicated and will be detailed elsewhere).We remark that Proca (scalar boson) stars can be con-tinuously connected to Kerr BHs via a larger family ofsolutions: Kerr BHs with Proca [15] (scalar [17]) hair,showing there is a general pattern and a general mecha-nism at work [18].
III. REFLECTION SPECTRUM
Broad iron lines are a common feature in the X-rayspectrum of both stellar-mass and supermassive BH can-didates. In the disk-corona model [19, 20], a BH is sur-rounded by a geometrically thin and optically thick ac-cretion disk. The disk emits like a blackbody locally anda multi-color blackbody when integrated radially. Thecorona is a hot ( ∼
100 keV), usually optically thin, elec-tron cloud. For instance, it may be the base of the jet ora cloud covering the BH or the inner part of the accre-tion disk, but the actual geometry is currently unknown.Due to inverse Compton scattering of thermal photonsfrom the disk off hot electrons in the corona, the lat-ter becomes an X-ray source with a power-law spectrum E − Γ , where Γ ≈ α line, which is atabout 6.4 keV in the case of neutral iron atoms and shiftsup to 6.97 keV in the case of H-like iron ions.Iron K α lines in the reflection spectrum of BHs arebroad and skewed as a result of relativistic effects(Doppler boosting, gravitational redshift, light bend-ing) occurring in the strong gravitational field of thesource. Assuming the Kerr metric, the analysis of theiron K α line can be used to measure the BH spin pa-rameter [21, 22]. Relaxing the Kerr BH hypothesis, thetechnique can probe the metric around the compact ob-ject [23, 24]. Actually one has to analyze the whole re-flection spectrum, not only the iron line, but most ofthe information about the spacetime metric in the stronggravity region is in the iron line and for this reason thetechnique is often referred to as the iron line method. Itis remarkable that, in the presence of high quality dataand the correct astrophysical model, this approach canbe a powerful tool to test the nature of BHs [25–27].The shape of the iron K α line is determined by themetric of the spacetime, the inclination angle of the diskwith respect to the line of sight of the distant observer i ,the geometry of the emitting region, and the emissivityprofile of the disk. The emission is usually assumed fromthe inner edge of the disk r in to some large outer radius r out , but the exact value of the latter is not importantbecause the emissivity is lower and lower as the radiusincreases. One usually tries to select the sources in whichthe inner edge of the disk is at the radius of the innermoststable circular orbit (ISCO). In the case of a corona witharbitrary geometry, it is common to model the emissivityprofile with a broken power-law, namely to assume thatthe emissivity scales as 1 /r q for r < r b and as 1 /r q for r > r b , where the emissivity indices q and q andthe breaking radius r b are three free parameters to bedetermined by the fit.In the case of the spacetimes of Proca stars, there is noISCO for co-rotating orbits, so the possible accretion diskmay either extend up to the center of the object or betruncated at some small radius. In our simulations, weassume the former scenario and we employ the followinglamppost-inspired emissivity profile [28]: I ∝ h ( r + h ) / , (3)where h is the height of the corona along the spin axisof the BH in the lamppost set-up and in our case wechoose h = 2 (in units in which 1 /µ = 1). The shapesof the expected iron lines in the reflection spectrum ofthe Proca stars with m = 0 (solutions 1-5) and m = 1(solutions 6-12) are shown, respectively, in the left andright panels in Fig. 2 for i = 45 ◦ . The calculations aredone with the code described in Refs. [29, 30]. IV. SIMULATIONS
We want now to address the question whether currentobservations of the reflection spectrum of astrophysicalBH candidates can rule out, or constrain, the possibil-ity that these objects are actually Proca stars. As an explorative study, we do not consider specific observa-tions. We instead follow the strategy already employedin Refs. [1, 2, 31, 32], which permits to get quickly arough estimate of current constraints. The key point isthat current observations are consistent with the Kerrmetric, in the sense that X-ray data are normally fittedwith reflection spectra computed in the Kerr metric andthe result is acceptable. We can thus simulate some ob-servations of the X-ray spectrum of Proca stars and thentry to fit the simulated data with a model calculated inthe Kerr background. If the fit is acceptable, we can saythat current data cannot rule out the Proca star solutionof that simulation. If the fit is bad, we can say that ourProca star solution cannot describe the metric aroundthe observed astrophysical BH candidates, because thereis currently no tension between observations and theoret-ical models.We simulate observations with XIS/Suzaku of abright BH binary. For simplicity, we model the spec-trum of the source with a power-law with photon in-dex Γ = 2 (representing the primary spectrum of thecorona) and a single iron line (the reflection spectrumof the disk). We assume typical parameters for a brightBH binary. The energy flux in the 0.7-10 keV range isabout 4 · erg/s/cm . The equivalent width of the ironline is about 200 eV. We assume that the exposure timeis 100 ks and the photon count turns out to be about3 · . For every Proca star solutions, we consider threeviewing angles, namely i = 20 ◦ , 45 ◦ , and 70 ◦ , but we findthat the final result is not very sensitive to i .We treat these simulations as real data. We useXSPEC and fit the simulated data with a power-lawand a Kerr iron line. The latter is modeled withRELLINE [33]. We have eight free parameters in the fit:the photon index of the power-law Γ, the normalizationof the power-law component, the spin of the BH a ∗ , theviewing angle i , the two indices q and q , the breakingradius r b , and the normalization of the iron line.The results of our simulations for the 12 Proca starsolutions can be summarized as follows. For m = 0, so-lution 1 cannot be fitted with a Kerr model, solution 2is marginally consistent, while solutions 3-5 can be wellfitted with a Kerr model (even if we obtain completelywrong estimates of some parameters). For m = 1, wehave a similar situation. Solution 6 cannot be well fittedwith a Kerr model, for solution 7 the fit is already accept-able, solutions 8-12 can be well fitted with a Kerr model.Fig. 3 shows the results for solutions 1, 2, 6, and 7 in thecase i = 45 ◦ . In every panel, the top plot shows the simu-lated data and the best fit (folded spectra). The bottompanel shows the ratio between the simulated data and thebest fit, which is the key-point in our simple analysis. Ifthe ratio is close to 1, in every region of the spectrumand within the error bars of the measurement, the model http://heasarc.gsfc.nasa.gov/docs/suzaku/ http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/index.html can well fit the data. This is clearly not the case for solu-tion 1 (top left panel) and solution 6 (bottom left panel)around 6 keV. In the case of solution 2 (top right panel)and solution 7 (bottom right panel), we see that at someenergies most of the data are above or below the line ofratio equal 1, but the error bars are large enough thatthe ratio 1 is included, and for solution 7 the fit is surelyacceptable. For a longer exposure time, the size of theerror bars would decrease and show clearly if the Kerrmodel cannot fit the data.As explained in the previous section, in our simulationsthe accretion disk extends untill the center at r = 0, be-cause there is no ISCO in these spacetimes for co-rotatingparticles. If we assume that the disk is truncated at somevery small radius, the resulting iron line is not very dif-ferent within the emissivity profile in Eq. (3) and ourconclusions are unchanged. If we move the inner edgeof the disk to larger radii, we cannot have the low en-ergy tail in the iron line profile and it is easier to ruleout these spacetimes. Our conclusions are thus based onthe simplest choice, which provides the most conservativebounds. V. CONCLUDING REMARKS
In this letter, we have extended previous work on KerrBHs with scalar hair [1] and boson stars [2] to the caseof the Proca star solutions found in Ref. [3]. We havestudied if current X-ray observations of the reflectionspectrum of stellar-mass BH candidates may be consis-tent with that expected for a Proca star or, otherwise,current observations can already rule out Proca stars asalternative to BHs.Our simulations show that current constraints on theexistence domain of Proca stars are weak. Still, we canrule out non-compact objects (solution 1 for m = 0 andsolution 6 for m = 1, corresponding also to the largestfrequencies w ), while more compact configurations canmimic Kerr BHs and have reflection spectra with broadiron lines. But observe that some of the most compactconfigurations are (or are likely , in the rotating case) un- stable, and hence may be ruled out on different grounds.Our constraints are weak for two reasons. First, theuncertainty in the emissivity profile limits the ability toperform precise measurements of the background met-ric. Second, with current X-ray missions the statisticsin the iron line is not good enough. These two pointsare not only true in the case of Proca stars, but for testsof the Kerr metric using X-ray reflection spectroscopyin general, and the results of this work confirm that atthe moment these are the two weak points of this ap-proach. Much stronger constraints should be possiblewith the next generation of X-ray missions. Timing mea-surements will be able to test the exact geometry of thecorona, which may permit to have a theoretical predic-tion of the emissivity profile. The much larger effectivearea of the next generation of X-ray missions will permitto have a sufficiently high photon number count to getprecise measurements of the spectra of the sources.To conclude, the analysis in this and our previous pa-pers [1, 2], establishes that with present and forthcomingastrophysical observations, the iron line technique can beused to set informative constraints on the domain of exis-tence of boson stars and their hairy BH counterparts. To-gether with other astrophysical information, such as thatobtained from gravitational lensing [34, 35] and quasi-periodic oscillations [36], these fairly simple alternativesto the Kerr BH paradigm, which occur within general rel-ativity albeit involving non-standard model matter, canundergo precision testing in the forthcoming years. ACKNOWLEDGMENTS
The work of T.S., M.Z., and C.B. was supportedby the NSFC (grants U1531117 and 11305038) and theThousand Young Talents Program. C.B. also acknowl-edges the support from the Alexander von HumboldtFoundation. C.A.R.H. and E.R. acknowledge fundingfrom the FCT-IF programme. This work was par-tially supported by the H2020-MSCA-RISE-2015 GrantNo. StronGrHEP-690904, and by the CIDMA projectUID/MAT/04106/2013. [1] Y. Ni, M. Zhou, A. Cardenas-Avendano, C. Bambi,C. A. R. Herdeiro and E. Radu, JCAP , 049 (2016)[arXiv:1606.04654 [gr-qc]].[2] Z. Cao, A. Cardenas-Avendano, M. Zhou, C. Bambi,C. A. R. Herdeiro and E. Radu, JCAP , 003 (2016)[arXiv:1609.00901 [gr-qc]].[3] R. Brito, V. Cardoso, C. A. R. Herdeiro and E. Radu,Phys. Lett. B , 291 (2016) [arXiv:1508.05395 [gr-qc]].[4] R. Narayan and J. E. McClintock, arXiv:1312.6698[astro-ph.HE].[5] C. Bambi, Rev. Mod. Phys. (in press) [arXiv:1509.03884[gr-qc]].[6] C. E. Rhoades and R. Ruffini, Phys. Rev. Lett. , 324 (1974).[7] E. Maoz, Astrophys. J. , L181 (1998) [astro-ph/9710309].[8] N. Yunes, K. Yagi and F. Pretorius, Phys. Rev. D ,084002 (2016) [arXiv:1603.08955 [gr-qc]].[9] D. J. Kaup, Phys. Rev. , 1331 (1968).[10] R. Ruffini and S. Bonazzola, Phys. Rev. , 1767(1969).[11] M. Colpi, S. L. Shapiro and I. Wasserman, Phys. Rev.Lett. , 2485 (1986).[12] C. A. R. Herdeiro, E. Radu and H. Rnarsson, Phys. Rev.D , no. 8, 084059 (2015) [arXiv:1509.02923 [gr-qc]].[13] F. E. Schunck and E. W. Mielke, Class. Quant. Grav. , P ho t on F l u x E obs (keV)m = 012345 P ho t on F l u x E obs (keV)m = 16789101112 FIG. 2. Expected shape of single iron lines in the reflection spectrum of Proca stars with m = 0 (solutions 1-5, left panel) and m = 1 (solutions 6-12, right panel). The viewing angle is i = 45 ◦ and the emissivity profile is ∝ h/ ( r + h ) / with h = 2. Seethe text for more details. FIG. 3. Results of our simulations and fits for Proca stars solution 1 (top left panel), solution 2 (top right panel), solution 6(bottom left panel), and solution 7 (bottom right panel). In every panel, the top plot shows the simulated data and the bestfit, while the bottom panel shows their ratio. See the text for more details.
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