Testing General Relativity with Accretion-Flow Imaging of Sgr A*
Tim Johannsen, Carlos Wang, Avery E. Broderick, Sheperd S. Doeleman, Vincent L. Fish, Abraham Loeb, Dimitrios Psaltis
aa r X i v : . [ a s t r o - ph . H E ] S e p Testing general relativity with accretion-flow imaging of Sgr A ∗ Tim Johannsen,
1, 2, 3
Carlos Wang, Avery E. Broderick,
1, 2
Sheperd S. Doeleman,
4, 5
Vincent L. Fish, Abraham Loeb, and Dimitrios Psaltis Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada Canadian Institute for Theoretical Astrophysics,University of Toronto, Toronto, Ontario M5S 3H8, Canada MIT Haystack Observatory, Westford, Massachusetts 01886, USA Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138, USA Department of Astronomy, Harvard University, 60 Garden Street, Cambridge, Massachusetts 02138, USA Astronomy Department, University of Arizona, 933 North Cherry Avenue, Tucson, Arizona 85721, USA
The Event Horizon Telescope is a global very-long baseline interferometer capable of probingpotential deviations from the Kerr metric, which is believed to provide the unique description ofastrophysical black holes. Here we report an updated constraint on the quadrupolar deviation ofSagittarius A ∗ within the context of a radiatively inefficient accretion flow model in a quasi-Kerrbackground. We also simulate near-future constraints obtainable by the forthcoming eight-stationarray and show that in this model already a one-day observation can measure the spin magnitude towithin 0 . . ◦ , the position angle to within 0 . ◦ , and the quadrupolardeviation to within 0 .
005 at 3 σ confidence. Thus, we are entering an era of high-precision stronggravity measurements. PACS numbers: 04.50.Kd,04.70.-s
The supermassive black holes Sgr A ∗ and in M87are the prime targets of the Event Horizon Telescope(EHT). These sources have already been observed witha three-station array, comprised by the James ClerkMaxwell Telescope and the Sub-Millimeter Array inHawaii (Hawaii), the Submillimeter Telescope Observa-tory in Arizona (SMT), and several dishes of the Com-bined Array for Research in Millimeter-wave Astronomy(CARMA) in California, which has resolved structureson scales of only 4 r S in Sagittarius A ∗ (Sgr A ∗ ) [1] and5 . r S in M87 [2], respectively. Here, r S ≡ r g ≡ GM/c is the Schwarzschild radius of a black hole with mass M ,and r g is its gravitational radius [3].According to the general-relativistic no-hair theorem,stationary, electrically neutral black holes in vacuum onlydepend on their masses M and spins J and are uniquelydescribed by the Kerr metric [4]. Mass and spin are thefirst two multipole moments of the Kerr metric, and allhigher-order moments can be expressed in terms of themby the relation M l + iS l = M ( ia ) l , where M l and S l arethe mass and current multipole moments, respectively,and a ≡ J/M is the spin parameter (see, e.g., Ref. [5]).General relativity has been well confirmed in theregime of weak gravitational fields [6], but still remainspractically untested in the strong-field regime foundaround compact objects [7]. It is possible to test theno-hair theorem using parametrically deformed Kerr-likespacetimes that depend on one or more free parametersin addition to mass and spin. Observations may then beused to measure the deviations. If none are detected, thecompact object is consistent with a Kerr black hole. If,however, nonzero deviations are measured, there are twopossible interpretations. If general relativity still holds, the object is not a black hole but, instead, another stablestellar configuration or some exotic object. Otherwise,the no-hair theorem would be falsified.By design, parametric Kerr-like spacetimes encompassmany theories of gravity at once and generally do notderive from the action of any particular theory. It isassumed, however, that particles move along geodesics.Tests of the no-hair theorem in a Kerr-like spacetimehave been suggested for gravitational-wave observationsof extreme mass-ratio inspirals [8] and electromagneticobservations of accretion flows surrounding black holes(e.g., [9, 10]). Other tests of the no-hair theorem includeelectromagnetic observations of pulsar black hole bina-ries [11, 12] and stars on orbits around Sgr A ∗ [12, 13],though these constitute weak-field probes.Here we employ a quasi-Kerr metric [14], which mod-ifies the quadrupole moment Q K ≡ M = − M a of theKerr metric according to the equation Q QK = − M ( a + ǫM ), where ǫ is a dimensionless parameter that mea-sures potential deviations from the Kerr metric. Thequasi-Kerr metric is of the form g QK µν = g K µν + ǫh µν , where g K µν is the Kerr metric and h µν is diagonal. An explicitexpression of this metric is given in Ref. [14]. Note thatthe expression for the quadrupole moment is exact forsufficiently small values of the spin and the parameter ǫ ,but it may only be approximate otherwise [15].A key objective of the EHT is to produce the first di-rect image of a black hole. These typically reveal a darkregion at the center, the so-called shadow [21]. The shapeof this shadow is exactly circular for a Schwarzschildblack hole and nearly circular for a Kerr black hole unlessits spin is very large and the inclination is high. How-ever, the shape of the shadow becomes asymmetric if theno-hair theorem is violated, e.g., for nonzero values ofthe parameter ǫ in the quasi-Kerr metric [9].Sgr A ∗ is the black hole with the largest angular crosssection in the sky. While several models for its ac-cretion flow exist [22], these typically fall within theradiatively inefficient accretion flow paradigm (RIAF).A recent analysis within the context of RIAFs foundthat images of accretion flows in the quasi-Kerr space-time differ significantly from those in a Kerr backgroundand, already, these may be grossly distinguished by earlyEHT data. Furthermore, measurements of the inclina-tion and spin position angle are robust to the inclusionof a quadrupolar deviation from the Kerr metric [23]. Inparticular, Ref. [23] obtained the 1 σ constraints on thespin magnitude a ∗ = 0 +0 . − . , inclination θ = 65 ◦ +21 ◦ − ◦ , andorientation ξ = 127 ◦ +17 ◦ − ◦ (up to a 180 ◦ degeneracy), whilethe deviation parameter ǫ remained unconstrained.In April 2009 and in March/April 2011–2013, addi-tional observations of Sgr A ∗ were carried out at 230 GHzusing the same three-station telescope array [24, 25].A comprehensive analysis of these observations togetherwith updated parameter estimates for Sgr A ∗ assuminga Kerr background can be found in Ref. [26]. Here, wefocus on the constraints on the quadrupolar deviation pa-rameter. Following the procedure described in Ref. [23]and allowing for closure phase shifts as in Ref. [26], weproduced an updated set of parameter estimates withinthe same parameter space [15] using the image library ofRef. [23] refined by an additional 12,501 images.Figure 1 shows the spin magnitude–inclination and thespin magnitude–quadrupolar deviation posterior proba-bility distributions, each marginalized over all other pa-rameters (spin orientation angle and, respectively, devi-ation parameter and inclination). As in Ref. [23], thespin magnitude correlates with both the inclination andthe deviation parameter, although it is unclear whetherthe latter correlation is still primarily determined by thelocation of the innermost stable circular orbit (ISCO).We obtain new constraints on the spin magni-tude a ∗ = 0 +0 . − . , inclination θ = 57 . ◦ +3 . ◦ − . ◦ , orienta-tion ξ = 156 ◦ +5 ◦ − ◦ , and deviation parameter ǫ = 1 . +0 . − . ,where we quote 1 σ errors on the respective posteriorprobability densities marginalized over all other param-eters [15, 27]. Formally this implies that Sgr A ∗ isconsistent with a Kerr black hole only at the 3 σ level.However, this constraint on the parameter ǫ is substan-tially biased by the restricted range of values of the spinand the quadrupolar deviation we consider affecting themarginalization process and, therefore, overestimatingthe magnitude of the deviation. A better measure is the2D probability distribution shown in Fig. 1, from whichit is clear that Sgr A ∗ is consistent with a Kerr blackhole well within the 2 σ level. Even though the maximumof this distribution is located at the edge of the parame-ter space, the confirmation of the Kerr nature of Sgr A ∗ ° ° ° ° ° ° ° a ∗ θ a ∗ ǫ FIG. 1. 1 σ , 2 σ , and 3 σ confidence contours of the poste-rior probability density as a function of (top) the spin mag-nitude a ∗ and inclination θ and (bottom) the spin magnitudeand quadrupolar deviation parameter ǫ , marginalized over allother quantities. The red dot in each panel denotes the max-imum of the respective 2D probability density and dashedwhite lines correspond to constant ISCO radii of (top) r = 6 r g and (bottom) r = 5 r g . The gray region is excluded. at this level should be unaffected by the considered val-ues of the spin and the deviation parameter, because the1 σ and 2 σ regions are very large. Therefore, we expectthat Sgr A ∗ remains consistent with a Kerr black holeat the 2 σ level even for larger values of the parameter ǫ . However, the quoted confidence intervals should beviewed with caution. Although our result implies thatSgr A ∗ is in mild tension with being a Kerr black hole, itis most likely dominated by systematic model uncertain-ties, which do not incorporate other effects such as thevertical structure and variability of the accretion disk,the plasma density and magnetic field strength in andabove the disk, and the presence of outflows.Our constraints on the parameters of Sgr A ∗ arebroadly consistent with the values given in Ref. [26] andimprove upon the constraints on the inclination and spinorientation of Ref. [23] by roughly a factor of four. Inaddition, the 180 ◦ degeneracy of the spin orientation isremoved. However, the constraint on the spin magnitudeis about 30% weaker than the constraint of Ref. [23] andthe spin magnitude is now unconstrained. This is in ac-cordance with the results of Refs. [28, 29] which foundthat the inclination and spin orientation can be inferredmuch more precisely from the visibility magnitudes andclosure phase data than the spin magnitude.In 2015, the three-station array Hawaii–SMT–CARMA was expanded to include the Atacama LargeMillimeter/submillimeter Array (ALMA) in Chile, theLarge Millimeter Telescope in Mexico, the South PoleTelescope (SPT), the Plateau de Bure interferometerin France, and the Pico Veleta Observatory in Spain.Thus, we also assess the prospects of measuring the spinmagnitude and position angle, the inclination, and thequadrupolar deviation parameter of Sgr A ∗ with an eight-station array in the near future. The sensitivity and res-olution of this enlarged array will be greatly increased,caused primarily by ALMA which will have a sensitivitythat is about 50 times greater than the sensitivity of thecurrent stations and the long baselines from the stationsin the northern hemisphere to the SPT. In addition, thisarray allows for the measurement of closure phases alongmany different telescope triangles, some of which dependvery sensitively on the parameters of Sgr A ∗ [30].To do this we simulate a single 24 h observing run at230 GHz using a library image with a ∗ = 0 . θ = 60 ◦ , ξ = 160 ◦ , and ǫ = 0, motivated by the results of Ref. [26].Simulated visibilities and closure phases are computedevery 10 min for all baselines comprised of telescopes forwhich Sgr A ∗ is above a zenith angle of 70 ◦ . Typical longbaseline observing periods are 2-4 h. For each station, weuse the system equivalent flux density at this frequencylisted in Ref. [31]. We assume a 4 GHz recording band-width and a 10 s atmospheric correlation time for allmeasurements. We further assume that the radio emis-sion experiences electron scatter broadening according tothe scattering law of Ref. [32].For our analysis, we created a new library of RIAFimages which consists of a total of 50,061 images withvalues of the spin magnitude 0 . ≤ a ∗ ≤ .
16, in-clination 59 . ◦ ≤ θ ≤ . ◦ , and deviation parameter − . ≤ ǫ ≤ . a ∗ = 0 . θ = 0 . ◦ , and ∆ ǫ = 0 . ξ in the range 159 . ◦ ≤ ξ ≤ . ◦ in steps of ∆ ξ = 0 . ◦ . All images were generated us- ° ° ° ° ° a ∗ θ −0.01−0.00500.0050.01 a ∗ ǫ FIG. 2. Simulated posterior probability density as a functionof (top) the spin magnitude a ∗ and inclination θ and (bottom)the spin magnitude and deviation parameter ǫ , marginalizedover all other quantities. Solid, dashed, and dotted white linesshow the 1 σ , 2 σ , and 3 σ confidence contours, respectively.The dashed magenta line denotes the location of the ISCOwith constant radius r = 5 . r g . ing the coefficients of the density and temperature of thethermal and nonthermal distribution of electrons fromfits of the radio spectral energy distribution of Sgr A ∗ obtained in Ref. [23]. For each library image, an associ-ated likelihood was constructed using the simulated datafollowing the procedure in Ref. [23].Figure 2 shows the spin magnitude–inclination and thespin magnitude–quadrupolar deviation posterior prob-ability distributions, respectively marginalized over allother quantities. The spin magnitude remains stronglycorrelated with the inclination, while the deviation pa-rameter ǫ is weakly correlated with the inclination. Nei-ther is correlated with the spin position angle. Allparameters in our simulation are tightly constrained: a ∗ = 0 . +0 . − . , θ = 60 . ◦ +0 . ◦ − . ◦ , ξ = 159 . ◦ ± . ◦ ,and ǫ = 0 ± . σ errors [15]. As weshow in the bottom panel of Fig. 2, the contours in the( a ∗ , ǫ ) plane are not exactly aligned with lines of constantISCO radius which suggests that this is not a fundamen-tal degeneracy (cf., the discussion in Ref. [33]).The reconstructed spacetime parameters are highlyprecise, despite adopting realistic station performance es-timates, and indicate the forthcoming capability of theEHT to probe deviations from general relativity. How-ever, these have been obtained within the context of aspecific astrophysical paradigm, placing an as yet poorlyunderstood prior on the analysis which will be investi-gated elsewhere. The effects of the chosen model and thevariability of the accretion flow are difficult to estimatequantitatively at this point, but these will most likelybe the dominant source of uncertainty. Such an assess-ment will require either increasingly parametrized modelsor further theoretical development of geometrically thickaccretion flows in order to account for the astrophysicaleffects neglected here. Nonetheless, our analysis demon-strates the expected dramatic improvement of the con-straints based on observations with a large EHT arraygiven one particular model.We have also neglected potential systematic errors as-sociated with the uncertain gain calibrations betweenlong and short baselines, estimated in currently reportedvisibility magnitudes to produce systematic variations ofup to 5% [24]. However, observations with the larger ar-ray will be able to mitigate these through substantiallyincreased sensitivity, redundant baselines, and the con-struction of closure amplitudes, defined for station quad-rangles, which are independent of station-specific gainestimates. Similarly, closure phases are by constructionindependent of station-specific phase errors, and will ben-efit from redundant triangles. Likewise, a scheduled in-crease of recording bandwidth will further increase thearray sensitivity by a factor √
8. At present, there alsoexist substantial uncertainties in our knowledge of the in-terstellar scattering law of Ref. [32], which, however, canbe rectified by additional observations and refined mod-eling [34]. Finally, refractive substructure along the lineof sight can cause stochastic variations of the image [35]which will average out if its blurring effect is sufficientlysmall [36] (cf., Ref. [28]). Therefore, we expect the overallimpact of these uncertainties on the simulated constraintsto be relatively small.Our results for the spin magnitude and deviation pa-rameter also depend on the mass and distance of Sgr A ∗ which affect the overall scale of the images and the spec-tral fits for the electron density and temperature. Cur-rent measurements of the mass and distance of Sgr A ∗ from near-infrared monitoring of stellar orbits close to theblack hole have relative errors on the order of 3% and 1%, respectively, which are, however, strongly correlated [37].These uncertainties will be further reduced by continuedmonitoring, by the expected improvement in astrometrywith the instrument GRAVITY for the Very Large Tele-scope Interferometer [38], in combination with EHT mea-surements of the shadow size of Sgr A ∗ [34, 39, 40], andcould reach a precision of ∼ ∗ to within 2%.T.J. was supported by a CITA National Fellowshipat the University of Waterloo and is supported in partby Perimeter Institute for Theoretical Physics. A.E.B.receives financial support from Perimeter Institute forTheoretical Physics and the Natural Sciences and Engi-neering Research Council of Canada through a DiscoveryGrant. Research at Perimeter Institute is supported bythe Government of Canada through Industry Canada andby the Province of Ontario through the Ministry of Re-search and Innovation. The Event Horizon Telescope issupported by grants from the National Science Founda-tion and from the Gordon and Betty Moore Foundation(Grant No. GBMF-3561). [1] S.S. Doeleman, J. Weintroub, A.E.E. Rogers, et al., Na-ture , 78 (2008).[2] S.S. Doeleman, V.L. Fish, D.E. Schenck, et al., Science , 355 (2012).[3] Hereafter we will set G = c = 1.[4] We expect astrophysical black holes to be electricallyneutral since any net charge would neutralize quickly.[5] M. Heusler, Black Hole Uniqueness Theorems (Cam-bridge University Press, Cambridge, England 1996).[6] C.M. Will, Living Rev. Rel. , 4 (2014).[7] D. Psaltis, Living Rev. Rel. , 9 (2008).[8] J.R. Gair, M. Vallisneri, S.L. Larson, and J.G. Baker,Living Rev. Rel. , 7 (2013).[9] T. Johannsen and D. Psaltis, Astrophys. J. , 446(2010).[10] T. Johannsen and D. Psaltis, Astrophys. J. , 11(2011); Astrophys. J. , 57 (2013); C. Bambi, Astro-phys. J. , 174 (2012); Phys. Rev. D , 023007 (2013);H. Krawczynski, Astrophys. J. , 133 (2012).[11] N. Wex and S.M. Kopeikin, Astrophys. J. , 388(1999); E. Pfahl and A. Loeb, Astrophys. J. , 253(2004); K. Liu, N. Wex, M. Kramer, J. M. Cordes, andT. J. W. Lazio, Astrophys. J. , 1 (2012).[12] D. Psaltis, N. Wex, and M. Kramer, Astrophys. J. ,121 (2016).[13] C.M. Will, Astrophys. J. , L25 (2008).[14] K. Glampedakis and S. Babak, Class. Quantum Grav. , 4167 (2006).[15] See Supplemental Material, which includes Refs. [16–20],for details on Kerr-like metrics, the considered parame-ter space, and the results of our analyses of current andfuture EHT data.[16] V. S. Manko and I. D. Novikov, Class. Quant. Grav. ,2477 (1992); N. A. Collins and S. A. Hughes, Phys. Rev. D , 124022 (2004); S.J. Vigeland and S.A. Hughes,Phys. Rev. D , 024030 (2010); T. Johannsen and D.Psaltis, Phys. Rev. D , 124015 (2011); S.J. Vigeland,N. Yunes, and L.C. Stein, Phys. Rev. D , 104027(2011).[17] T. Johannsen, Phys. Rev. D , 044002 (2013).[18] J.B. Hartle, Astrophys. J. , 807 (1968).[19] T. Johannsen, Class. Quantum Grav. , 113001 (2016).[20] T. Johannsen, Phys. Rev. D , 124017 (2013).[21] J.M. Bardeen in Black Holes , Gordon and Breach (1973);H. Falcke, F. Melia, and E. Agol, Astrophys. J. , L13(2000).[22] R. Narayan, R. Mahadevan, J.E. Grindlay, R.G.Popham, and C. Gammie, Astrophys. J. , 554 (1998);R.D. Blandford and M.C. Begelman, Mon. Not. R. As-tron. Soc. , L1 (1999); H. Falcke and P.L. Biermann,Astron. Astrophys. , 49 (1999); F. ¨Ozel, D. Psaltis,and R. Narayan, Astrophys. J. , 234 (2000); F. Yuan,E. Quataert, and R. Narayan, Astrophys. J. , 301(2003).[23] A.E. Broderick, T. Johannsen, A. Loeb, and D. Psaltis,Astrophys. J. , 7 (2014).[24] V.L. Fish, S.S. Doeleman, C. Beaudoin, et al., Astrophys.J. , L36 (2011).[25] V.L. Fish, M.D. Johnson, S.S. Doeleman, et al., Astro-phys. J. , 90 (2016).[26] A.E. Broderick, V.L. Fish, M.D. Johnson, et al., Astro-phys. J. , 137 (2016).[27] Here, a ∗ ≡ a/M is the dimensionless spin parameter.[28] L. Medeiros, C.-k. Chan, F. ¨Ozel, et al., submitted toAstrophys. J., arXiv:1601.06799.[29] J. Kim, D.P. Marrone, C.-k. Chan, et al., submitted toAstrophys. J., arXiv:1602.00692; R. Fraga-Encinas, M.Mo´scibrodzka, C. Brinkerink, and H. Falcke, Astron. As-trophys. , A57 (2016).[30] A.E. Broderick, V.L. Fish, S.S. Doeleman, and A. Loeb,Astrophys. J. , 38 (2011).[31] S.S. Doeleman, V.L. Fish, A.E. Broderick, A. Loeb, andA.E.E. Rogers, Astrophys. J. , 59; R.-S. Lu, A.E.Broderick, F. Baron, J.D. Monnier, V.L. Fish, S.S. Doele-man, and V. Pankratius, Astrophys. J. , 120 (2014).[32] G.C. Bower, W.M. Goss, H. Falcke, D.C. Backer, and Y.Lithwick, Astrophys. J. , L127 (2006).[33] T. Johannsen, Class. Quantum Grav. , 124001 (2016).[34] D. Psaltis, F. ¨Ozel, C.-K. Chan, and D.P. Marrone, As-trophys. J. , 115 (2015).[35] V.L. Fish, M.D. Johnson, R.-S. Lu, et al., Astrophys. J. , 134 (2014); M.D. Johnson and C.R. Gwinn, Astro-phys. J. , 180 (2015).[36] R.-S. Lu, F. Roelofs, V.L. Fish, et al., Astrophys. J.
173 (2016); A.A. Chael, M.D. Johnson, R. Narayan,S.S. Doeleman, J.F.C. Wardle, and K.L. Bouman,arXiv:1605.06156.[37] A.M. Ghez, S. Salim, N.N. Weinberg, et al., Astrophys. J. , 1044 (2008); S. Gillessen, F. Eisenhauer, S. Trippe,et al., Astrophys. J. , 1075 (2009); L. Meyer, A.M.Ghez, R. Sch¨odel, S. Yelda, A. Boehle, J.R. Lu, T. Do,M.R. Morris, E.E. Becklin, and K. Matthews, Science , 84 (2012); T. Do, G.D. Martinez, S. Yelda, A.M.Ghez, J. Bullock, M. Kaplinghat, J.R. Lu, A.G.H. Peter,and K. Phifer, Astrophys. J. , L6 (2013); S. Chat-zopoulos, T.K. Fritz, O. Gerhard, S. Gillessen, C. Wegg, R. Genzel, and O. Pfuhl, Mon. Not. R. Astron. Soc. ,948 (2015).[38] F. Eisenhauer, G. Perrin, W. Brandner, et al., The Mes-senger , 16 (2011).[39] T. Johannsen, D. Psaltis, S. Gillessen, D.P. Marrone, F.¨Ozel, S.S Doeleman, and V.L. Fish, Astrophys. J. ,30 (2012).[40] T. Johannsen, A.E. Broderick, P.M. Plewa, et al., Phys.Rev. Lett. , 031101 (2016).[41] N.N. Weinberg, M. Milosavljevi´c, and A.M. Ghez, Astro-phys. J. , 878 (2005).[42] M.J. Reid, K.M. Menten, A. Brunthaler, et al., Astro-phys. J. , 130 (2014).
Supplemental Material
Context of other Kerr-like metrics —
Several Kerr-likemetrics have been proposed to date (e.g., [14, 16, 17]).The quasi-Kerr metric derives from the Hartle-Thornemetric [18] which was originally developed for the de-scription of neutron stars. The quasi-Kerr metric is asolution of the vacuum Einstein equations for spins thatsatisfy | a ∗ | ≪
1, provided ǫ is small. Here, however, wetreat the quasi-Kerr metric as an “exact” metric as dis-cussed in Ref. [20] and study the impact of a deformedquadrupole moment on black hole accretion flows. Thus,we neither require the spin a ∗ nor the deviation param-eter ǫ to be small.As shown in Ref. [20], the quasi-Kerr metric actuallyharbors a naked singularity as well as pathological regionsof space around this singularity where closed timelikecurves exist and Lorentzian symmetry is violated. There-fore, as in Ref. [23], we impose a cutoff radius at r = 3 r g ,which encloses all unphysical regions, and we considerall photons and matter particles that pass through thisradius “captured,” i.e., they no longer contribute to oursimulation. With this setup, the quasi-Kerr metric ef-fectively describes a black hole. In addition, we onlyconsider values of the spin and the parameter ǫ for whichthe ISCO lies at a radius r ≥ r g . These restrictionsdefine the excluded region shown as the gray region inthe bottom panels of Figs. 1 and 2 and ensure that thequasi-Kerr part of the metric is always much smaller inmagnitude than the Kerr part. Consequently, our sim-ulation actually tends to underestimate the effect of thedeviation from the Kerr metric, because its impact wouldbe the strongest at small radii which we partially exclude.The quasi-Kerr metric has the advantage of being ofa particularly simple form shortening the computationaltime required for extensive parameter studies (thoughthis is not a critical limitation). In addition, while otherKerr-like metrics such as the one of Ref. [17] are muchbroader in scope and have physical properties that arebetter suited for tests of the no-hair theorem (see thediscussion in Ref. [17]), the use of the quasi-Kerr metricallows us to establish direct contact between our results ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ξθ −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 130 ° ° ° ° ° ° ° ǫθ −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 180 ° ° ° ° ° ° ° ° ǫξ ° ° ° ° ° ° ° ° ξθ −0.01 −0.005 0 0.005 0.0159.8 ° ° ° ° ° ǫθ −0.01 −0.005 0 0.005 0.01159.9 ° ° ° ǫξ FIG. 3. 1 σ , 2 σ , and 3 σ confidence contours of the posterior probability density as a function of (left panels) inclination θ andthe spin orientation ξ , (center panels) the inclination and the deviation parameter ǫ , and (right panels) the spin orientationand the deviation parameter, marginalized over all other quantities. The top row panels correspond to the current constraintson these parameters from existing EHT data, while the bottom row panels correspond to our simulation of near-future EHTobservations. The red dot in each panel denotes the maximum of the respective 2D probability density. and the analysis of early EHT data of Ref. [23] and high-lights the tremendous improvement in precision achiev-able with larger EHT arrays. On the other hand, themetric of Ref. [17] can more easily accommodate largedeviations from the Kerr metric which are favored in ourcurrent analysis and can be mapped to known black-holesolutions in certain alternative theories of gravity (seeRefs. [19, 40]). See Ref. [19] for a review on Kerr-likemetrics and tests of the no-hair theorem with electro-magnetic observations of Sgr A ∗ .In a different interpretation, the free parameter ǫ canalso be regarded as a measure of the underlying system-atic uncertainties in the measurement (see the discussionin Ref. [20]). Comparing the results of our analysis ofthe current EHT data with the results of Ref. [26] whichare based on the same RIAF model but assume the Kerrmetric instead (i.e., ǫ = 0), the additional degree of free-dom in terms of the parameter ǫ points to the presenceof substantial systematic uncertainties (linked to, e.g.,variability, the morphology of the accretion disk, and po-tential outows; see, also, the discussion in Ref. [26]) whichneed to be incorporated. While our analysis yields robust results for the inclination and spin orientation, the spinmagnitude is unconstrained (cf., Refs. [28, 29]). Parameter space —
Our analysis of the current dataconsiders the same parameter space as Ref. [23] with up-per bounds on the spin magnitude a ∗ ≤ . ǫ ≤
1. These are conventions, chosen suchthat the quasi-Kerr part of the metric is always muchsmaller in magnitude compared to the Kerr part of themetric. A spin-dependent lower bound on the parameter ǫ is determined by the properties of the quasi-Kerr metricat small radii such that the ISCO is located at a radius r ≥ r g for any pair ( a ∗ , ǫ ). Since the RIAF model isprobably not well defined for counter-rotating disks [22],we also require a ∗ ≥