Testing conformal gravity with astrophysical black holes
TTesting conformal gravity with astrophysical black holes
Cosimo Bambi,
1, 2, ∗ Zheng Cao, and Leonardo Modesto † 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 Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China (Dated: March 8, 2017)Weyl conformal symmetry can solve the problem the spacetime singularities present in Einstein’sgravity. In a recent paper, two of us have found a singularity-free rotating black hole solution inconformal gravity. In addition to the mass M and the spin angular momentum J of the black hole,the new solution has a new parameter, L , which here we consider to be proportional to the blackhole mass. Since the solution is conformally equivalent to the Kerr metric, photon trajectories areunchanged, while the structure of an accretion disk around a black hole is affected by the valueof the parameter L . In this paper, we show that X-ray data of astrophysical black holes require L/M < . I. INTRODUCTION
Einstein’s theory of general relativity has been able topass a large number of tests, and there is currently noclear evidence of disagreement between theoretical pre-dictions and observational data [1]. However, we knowthat the theory breaks down in some extreme conditions.In particular, there are physically relevant solutions withspacetime singularities, where predictability is lost andstandard physics cannot be applied.The resolution of spacetime singularities in Einstein’sgravity is an outstanding and longstanding problem. Dif-ferent authors have explored different approaches. It ispossible that the resolution of spacetime singularities inEinstein’s gravity is related to the yet unknown theoryof quantum gravity. Among the many proposals presentin the literature, in this paper we are interested in thefamily of conformal theories of gravity [2–6].In conformal gravity, the theory is invariant under aconformal transformation of the metric tensor; that is, g µν → g ∗ µν = Ω g µν , (1)where Ω = Ω( x ) is a function of the point of the space-time. Einstein’s gravity is not conformally invariant, butit can be made conformally invariant, for instance by in-troducing an auxiliary field. There are many possiblerealizations in the literature [2–6]. Examples of confor-mal theories of gravity in four dimensions are L = a C µνρσ C µνρσ + b R µνρσ R µνρσ , L = φ R + 6 g µν ( ∂ µ φ )( ∂ ν φ ) , where C µνρσ is the Weyl tensor, R µνρσ is the Riemanntensor, a and b are constants, and φ is an auxiliary scalarfield (dilaton). In our case, we are not interested in aparticular model, but we just require the theory to be in-variant under the transformation (1). While the theory is ∗ [email protected] † [email protected] invariant under conformal transformations, a Higgs-likemechanism may choose one of the metric as the “phys-ical” solution to describe the spacetime. The Universeindeed does not appear to be conformally invariant, butthere is evidence of scale invariance in the early Uni-verse and in many other physical phenomena, like phasetransitions, etc. Moreover, we tend to believe that thephysics very close to the black hole may be described by aquantum gravity theory in its conformal invariant phasebecause the Hawking temperature gets a huge blueshiftnear the event horizon. We could thus expect large devi-ations from the Kerr metric because the energy becomesvery large (trans Planckian) and the conformal invarianceis restored while the vacuum becomes degenerate.Conformal gravity can solve the problem of spacetimesingularities by finding a suitable conformal transforma-tion Ω that removes the singularity and by interpretingthe metric g ∗ µν as the physical metric of the spacetime.In Ref. [7], we found a singularity-free rotating black holesolution conformally equivalent to the Kerr metric (seeRef. [8] on how conformal invariance is preserved at thequantum level). In Boyer-Lindquist coordinates, the lineelement reads ds = (cid:18) L Σ (cid:19) ds , (2)where L > r + a cos θ , and ds is the line element of the Kerr metric ds = − (cid:18) − M r Σ (cid:19) dt − M ar sin θ Σ dtdφ (3)+ Σ∆ dr + Σ dθ + (cid:18) r + a + 2 M a r sin θ Σ (cid:19) sin θdφ ,a = J/M is the rotational parameter (the dimensionlessspin parameter is a ∗ = a/M ) and ∆ = r − M r + a . Let us note that Eq. (2) is an exact rotating blackhole solution in a large family of theories. Black holesin theoretically-motivated alternative theories of gravityare usually known in the non-rotating case, sometimesin the slow-rotation approximation, but there are only a a r X i v : . [ g r- q c ] M a r r I S C O aL = 0.0L = 0.5L = 1.0L = 1.3L = 1.5L = 1.6 FIG. 1. ISCO radius r ISCO as a function of the specific spin a for different values of the parameter L . r ISCO , a , and L inunits in which M = 1. few examples in the literature of exact solutions for anyvalue of the spin.Black holes in theories beyond Einstein’s gravity havebeen extensively investigated in the past 10-20 years.They may have different theoretical (e.g. thermodynam-ics stability, uniqueness of solutions, entropy, topologyof the horizon, etc.) and observational properties withrespect to the Kerr black holes of Einstein’s gravity, sothat well-known results valid in the standard theory maynot hold. See, for instance, [9–13] and reference therein.In this work, we want to “test” the metric in Eq. (2)and constrain the parameter L . At the moment thereare no indications from the theory about the value of L ,which may thus be expected either of the order of 1 (inPlanck units) or of the order of M . There are no otherscales. If L is of the order of the Planck length, there isno way to get an estimate of its value with the techniquediscussed in this paper. We thus consider the second,more favorable, case, in which L can be of the order of M . There are indeed scenarios in the literature in whichone can expect macroscopic deviations at the scale ofthe horizon [14–17]. As we will show below within asimple analysis, current X-ray data of astrophysical blackholes require L/M < .
2. Since the metric in Eq. (2)is conformally equivalent to the Kerr solution, photontrajectories are the same as in the Kerr metric. However,the structure of the accretion disk changes. In particular,the value of L alters the innermost stable circular orbit(ISCO), which has a strong impact on current techniqueslike the continuum-fitting and the iron line methods. II. ACCRETION DISK
There are a number of proposals to probe thespacetime metric around astrophysical black holes withelectromagnetic radiation [13, 18]. At present, thecontinuum-fitting and the iron line methods can providesome general constraints, while other techniques are not yet mature to test fundamental physics or the necessaryobservational data are not yet available.The continuum-fitting and the iron line methods as-sume that the accretion disk is described by the Novikov-Thorne model [19, 20], which is the standard frameworkfor geometrically thin and optically thick accretion disksaround black holes. The disk is in the equatorial plane,perpendicular to the black hole spin. The particles of thegas follow nearly geodesic equatorial circular orbits. Acrucial ingredient of the model is that the inner edge ofthe disk is at the ISCO radius. This assumption playsa fundamental role in the continuum-fitting and the ironline methods and is supported by the observed stabilityof the position of the inner edge of the disk [21].The ISCO radius for a generic stationary, axisymmet-ric, and asymptotically flat spacetime can be computedas follows (see, for instance, Appendix B in [22] for moredetails). We write the line element in the canonical form,namely ds = g tt dt + g tφ dtdφ + g rr dr + g θθ dθ + g φφ dφ , (4)where the metric coefficients are independent of t and φ .We have thus two constants of motion, namely the spe-cific energy E and the axial component of the specific an-gular momentum L z . We write the t - and φ -componentsof the 4-velocity of a particle in terms of E , L z , and themetric coefficients. From the conservation of the rest-mass g µν ˙ x µ ˙ x ν = −
1, we write g rr ˙ r + g θθ ˙ θ = V eff ( r, θ ) , (5)where the dot ˙ indicates the derivative with respect tothe particle proper time and the effective potential V eff is V eff = E g φφ + 2 EL z g tφ + L z g tt g tφ − g tt g φφ − . (6)Circular orbits in the equatorial plane have V eff = ∂ r V eff = ∂ θ V eff = 0. The orbits are stable under smallperturbations if ∂ r V eff ≤ ∂ θ V eff ≤
0. The ISCOradius is found when either ∂ r V eff or ∂ θ V eff vanish.Fig. 1 shows the ISCO radius r ISCO as a function ofthe rotational parameter a for different values of L . Thevalue of the radial coordinate has not a direct physicalmeaning, as it depends on the choice of the coordinates,which is arbitrary. However, from Fig. 1 we see that,as the value of L increases, the minimum value of r ISCO increases too. We can already anticipate that this impliesthat very broad iron lines – as we observe in the X-raydata of some black holes – are not possible for large L ,because the gravitational redshift would be too weak. III. IRON K α LINE
Now we want to take a step forward and calculate theiron K α line that can be expected in the reflection spec-trum of an accretion disk around a black hole with L > P ho t on F l u x E obs (keV)a = 0L = 0.0L = 0.5L = 1.0L = 1.3L = 1.5L = 1.6 P ho t on F l u x E obs (keV)a = 0.95L = 0.0L = 0.5L = 1.0L = 1.3L = 1.5L = 1.6 FIG. 2. Iron line shapes in the case of non-rotating black hole solutions ( a ∗ = 0, left panel) and fast-rotating black hole solutions( a ∗ = 0 .
95, right panel) for different values of the conformal length L . The viewing angle in these simulations is i = 45 ◦ andthe emissivity profile of the disk is assumed ∝ /r . Within the disk-corona model [23, 24], we have a thinaccretion disk surrounding a black hole. The corona isa hot ( ∼
100 keV), usually optically thin, cloud. Thecorona may be the base of a jet, a sort of atmosphereabove the accretion disk, etc. The actual geometry iscurrently unknown. Due to inverse Compton scatteringof thermal photons from the disk from free electrons inthe corona, the latter becomes an X-ray source with apower-law spectrum. A fraction of these X-ray photonscan illuminate the disk, producing the so-called reflectionspectrum with some fluorescent emission lines [25]. Themost prominent line is usually the iron K α one, whichis at 6.4 keV in the case of neutral iron and shifts up to6.97 keV in the case of H-like iron ions.The iron K α line is very narrow in the rest-frame ofthe emitter, but it can appear broad and skewed in thespectrum of a black hole as the result of relativistic effects(Doppler boosting, gravitational redshift, light bending)occurring in the strong gravity region. If we assume theKerr metric, the analysis of the iron line can provide anestimate of the black hole spin [26, 27]. If we relax theKerr black hole hypothesis, this technique can be used toconstrain possible deviations from the Kerr solution. Letus note that, in the presence of high quality data and ofthe correct astrophysical model, the iron line method canbe a powerful tool to test the spacetime metric aroundastrophysical black holes [28–30]. Actually, one has tofit the whole reflection spectrum, not just the iron line,but the latter is the most prominent feature and, in apreliminary analysis like the present work, we can restrictour attention to the iron K α line only.Fig. 2 shows the expected iron line profiles in the re-flection spectrum of a black hole with, respectively, thespin parameter a ∗ = 0 (left panel) and 0.95 (right panel)and different values of L . The calculations have beendone with the code described in Refs. [31, 32]. The case L = 0 corresponds to the Kerr metric. In these calcu-lations, we have employed the emissivity profile ∝ /r , which corresponds to the case of the Newtonian limit (nolight bending) at large radii for a point-like corona justabove the black hole. However, for our considerations itdoes not play an important role, because the choice ofthe intensity profile can only alter the shape of the ironline. It cannot determine the photon energy detected atinfinity from a specific point of the disk. From Fig. 2, wesee that, as L increases, the observed iron line profile be-comes less broad. This is true even for the non-rotatingcase (left panel) although the ISCO radius may actuallybe smaller than the case with L = 0 (see Fig. 1). X-raydata of some black holes clearly show very broad iron linethat can extend to low energies. This would not be pos-sible with a sufficiently high value of L , and this permitsus to constrain the value of L . IV. SIMULATIONS
Let us now be more quantitative and get a constrainton the value of L from the iron line. A rough estimateof the maximum value of L allowed by current X-raydata can be obtained quite easily following the methodalready employed in Refs. [33–36]. We know that currentX-ray data are consistent with the Kerr metric, in thesense that observations are fitted with theoretical Kerrmodel and we obtain acceptable fits. We also know thatsome iron line are very broad and, if interpreted withinthe Kerr metric, it means that the spin parameter ofthe black hole is close to 1. For instance, assuming theKerr metric, for Cygnus X-1 we have the measurement a ∗ = 0 . +0 . − . [37], for LMC X-1 a ∗ = 0 . +0 . − . [38] ,for GX 339-4 a ∗ = 0 . +0 . − . [39].As done in Refs. [33–36] for other scenarios, we simu-late some observations taking into account the responseof the instrument, the background noise, and the intrin-sic Poisson noise of the source. We employ the typicalparameters for a bright stellar-mass black hole in a bi-nary. We model the X-ray spectrum of the source with apower-law ∝ E − Γ with Γ = 2 (the primary spectrum ofthe corona) and a single iron line (the reflection spectrumfrom the disk). This is a simple model, but it should cap-ture the main features of the new metric and is enoughfor getting a rough constraint on L . We assume that theenergy flux in the 3-10 keV range is 6 · − erg/s/cm and that the equivalent width of the iron line is around200 eV. We simulate observation with NuSTAR , assum-ing that the exposure time is 100 ks.We have simulated a number of observations for a ∗ =0 .
95 and different values of the parameters L , of the view-ing angle i , and with different emissivity profiles. Thesimulations have been treated as real data and fittedwith XSPEC , employing the model RELLINE for theiron line [41]. If L is not very large, the fits are usuallyacceptable, in the sense that there are not unresolvedfeatures and the minimum of the reduced χ is not muchhigher than 1. If we require that the analysis of our sim-ulations should provide spin measurements a ∗ > .
9, wefind that a conservative constraint for L is L/M < . V. CONCLUDING REMARKS
In this paper, we have discussed how to test conformalgravity with observational data of astrophysical blackholes. In particular, we have considered the singularity-free rotating black hole solution in Eq. (2) and obtaineda rough constraint on the parameter L . The latter canchange the structure of the accretion disk around blackholes and, in particular, can alter the ISCO radius, aquantity that can have a strong impact on the X-rayspectrum of black holes. Within a very simple analysis,we have discussed the expected iron line in the reflectionspectrum of these spacetimes and shown that current ob-servations require L/M < .
2. For higher values of L ,it is not possible to reproduce the extended low energytail in the iron line as observed in the spectra of severalsources.We would like to remark that our analysis is quite sim-ple, but the constraint L/M < . XSPEC is an X-ray spectral-fitting software com-monly used in X-ray astronomy. See [40] andhttp://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/index.htmlfor more details. disk. Our result does not depend on the intensity profile,because the key-point of the constraint
L/M < . ∼ L/M we cannot have very redshifted photons at any radius ofthe accretion disk. The photon redshift is only deter-mined by the background metric, and it is not possibleto increase the redshift factor of the photons by chang-ing the astrophysical model. A more detailed analysiswith the more sophisticated model presented in Ref. [42]can study real data of specific sources and get strongerconstraints. We postpone such an analysis to a futurework.
ACKNOWLEDGMENTS
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