Thin Accretion Disk Signatures in Dynamical Chernsimons-Modified Gravity Tiberiu Harko, Zoltán Kovács, Francisco S N Lobo

Home , Accretion disk

Thin accretion disk signatures in dynamical ChernSimons-modified gravity Tiberiu Harko, Zoltán Kovács, Francisco S N Lobo

To cite this version:

Tiberiu Harko, Zoltán Kovács, Francisco S N Lobo. Thin accretion disk signatures in dynamical ChernSimons-modified gravity. Classical and Quantum Gravity, IOP Publishing, 2010, 27(10), pp.105010. 10.1088/0264-9381/27/10/105010. hal-00597854

HAL Id: hal-00597854 https://hal.archives-ouvertes.fr/hal-00597854 Submitted on 2 Jun 2011

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Confidential: not for distribution. Submitted to IOP Publishing for peer review 8 March 2010

Thin accretion disk signatures in dynamical Chern-Simons modiﬁed gravity

Tiberiu Harko∗ and Zolt´an Kov´acs† Department of Physics and Center for Theoretical and Computational Physics, The University of Hong Kong, Pok Fu Lam Road, Hong Kong

Francisco S. N. Lobo‡ Centro de F´ısica Teórica e Computacional, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Ed. C8 1749-016 Lisboa, Portugal (Dated: March 8, 2010) ApromisingextensionofgeneralrelativityisChern-Simons(CS)modifiedgravity,inwhichthe Einstein-Hilbert action is modified by adding a parity-violating CS term, which couples to gravity via a scalar field. In this work, we consider the interesting, yet relatively unexplored, dynamical formulation of CS modified gravity, where the CS coupling field is treated as a dynamical field, endowed with its own stress-energy tensor and evolution equation. We consider the possibility of observationally testing dynamical CS modified gravity by using the accretion disk properties around slowly-rotating black holes. The energy flux, temperature distribution, the emission spectrum as well as the energy conversion efficiency are obtained, and compared to the standard general relativistic Kerr solution. It is shown that the Kerr black hole provide a more efficient engine for the transformation of the energy of the accreting mass into radiation than their slowly-rotating counterparts in CS modified gravity. Specific signatures appear in the electromagnetic spectrum, thus leading to the possibility of directly testing CS modified gravity by using astrophysical observations of the emission spectra from accretion disks.

PACS numbers: 04.50.Kd, 04.70.Bw, 97.10.Gz

I. INTRODUCTION date, has considered the nondynamical formulation [7– 10], whereas the dynamical formulation remains mostly unexplored territory. Recently, modified theories of gravity have received a Relative to rotating black hole spacetimes, several solu- considerable amount of attention mainly motivated by tions in the nondynamical formulation were found in CS the problems of dark energy (see [1] for reviews) and modified gravity [8–10]. The first solutions were found dark matter [2], and from quantum gravity. A promis- by Alexander and Yunes [8, 9], using a far-field approxi- ing extension of general relativity is Chern-Simons (CS) mation (where the field point distance is considered to be modified gravity [3–5], in which the Einstein-Hilbert ac- much larger than the black hole mass). The second ro- tion is modified by adding a parity-violating CS term, tating black hole solution was found by Konno et al [10], which couples to gravity via a scalar field. It is interest- using a small slow-rotation approximation, where the ing to note that the CS correction introduces a means to spin angular momentum is assumed to be much smaller enhance parity violation through a pure curvature term, than the black hole mass. However, it is interesting to as opposed to through the matter term, as is usually con- note that recently, using the dynamical formulation of sidered in general relativity. In fact, CS modified grav- CS modified gravity, spinning black hole solutions in the ity can be obtained explicitly from superstring theory, slow-rotation approximation have been obtained [11, 12]. where the CS term in the Lagrangian density is essen- tial due to the Green-Schwarz anomaly-canceling mech- An interesting feature of CS modified gravity is that it anism, upon four-dimensional compactification [6]. Two has a characteristic observational signature, which could formulations of CS modified gravity exist as independent allow to discriminate an effect of this theory from other theories, namely, the nondynamical formulation and the phenomena. However, most of the tests of CS modified dynamical formulation (see [5] for an excellent recent re- gravity to date have been performed with astrophysical view). In the former, the CS scalar is an aprioripre- observations and concern the non-dynamical framework. scribed function, where its effective evolution equation In particular, it was found that the CS modified theory reduces to a differential constraint on the space of allowed predicts an anomalous precession effect [14], which was solutions; in the latter, the CS is treated as a dynami- tested [15] with LAGEOS [16]. Another constraint on the cal field, possessing an effective stress-energy tensor and non-dynamical theory was proposed in [17], where it was an evolution equation. The majority of the work, up to considered that the CS correction could be used to ex- plain the flat rotation curves of galaxies. However, in [18] aboundwasplacedonthenon-dynamicalmodelwitha canonical CS scalar that is eleven orders of magnitude ∗Electronic address: [email protected] stronger than the Solar System one, using double binary †Electronic address: [email protected] pulsar data. Recently, using the dynamical formulation ‡Electronic address: [email protected] of CS modified gravity, a stringent constraint was placed 2 on the coupling parameter associated to the dynamical The first term is the standard Einstein-Hilbert action coupling of the scalar field [19]. √ Z 4 In this work, we further extend the constraints placed SEH = κ d x −gR , (2) on the dynamical formulation of CS gravity by using the observational signatures of thin disk properties around where κ−1 =16πG and R is the Ricci scalar. The second rotating black holes. In the context of stationary axisym- term defined as metric spacetimes, the mass accretion around rotating α √ black holes was studied in general relativity for the first S = Z d4x −gϑ ∗RR, (3) time in [20], by extending the theory of non-relativistic CS 4 accretion [21]. The radiation emitted by the disk surface was also studied under the assumption that black is the Chern-Simons correction; the third term body radiation would emerge from the disk in thermo- √ β Z 4 µν dynamical equilibrium [22, 23]. More recently, the emis- Sϑ = − d x −g [g (∇µϑ)(∇ν ϑ)+2V (ϑ)] , (4) 2 sivity properties of the accretion disks were investigated for exotic central objects, such as wormholes [24], and is the scalar field term. The matter action is given by non-rotating or rotating quark, boson or fermion stars, √ brane-world black holes or gravastars [25–34]. Z 4 Smat = d x −gLmat , (5) Thus, it is the purpose of the present paper to study the thin accretion disk models for slowly-rotating black where L the matter Lagrangian. holes in the dynamical formulation of CS modified the- mat The parameters α and β are dimensional coupling con- ories of gravity, and carry out an analysis of the prop- stants; the CS coupling field, ϑ,isafunctionofspace- erties of the radiation emerging from the surface of the time that parameterizes deformations from GR [12]; ∇ disk. As compared to the standard general relativistic µ is the covariant derivative associated with the metric ten- case, significant differences appear in the energy flux and sor g ;andthequantity∗RR is the Pontryagin density electromagnetic spectrum for CS slowly-rotating black µν defined as holes, thus leading to the possibility of directly testing CS modified gravity by using astrophysical observations ∗ ∗ τ µν σ RR = R σ R τµν , (6) of the emission spectra from accretion disks. ∗ τ µν The present paper is organized as follows. In Sec. II, we where the dual Riemann tensor is given by R σ = 1 µναβ τ µναβ review the dynamical formulation of CS modified grav- 2 R σαβ,with the 4-dimensional Levi-Civita ity, and present the Yunes-Pretorius (YP) slowly-rotating tensor. solution found in [12]. In Sec. III, we review the formal- Varying the action S with respect to the metric gµν ism and the physical properties of the thin disk accretion one obtains the gravitational field equation given by onto compact objects, for stationary axisymmetric spacetimes. In Sec. IV, we analyze the basic properties of mat- α 1 mat ϑ Gµν + Cµν = Tµν + Tµν , (7) ter forming a thin accretion disk in slowly-rotating black κ 2κ hole spacetimes in CS modified gravity. We discuss and where Gµν is the Einstein tensor, and Cµν is the cotton conclude our results in Sec. V. Throughout this work, tensor defined as we use a system of units so that c = G =¯h = kB =1, µν σαβ(µ ν) ∗ α(µν)σ where kB is Boltzmann’s constant. C = ∇σϑ ∇β R α + ∇σ∇αϑ R . (8) The total stress-energy tensor is split into the matter µν µν term Tmat,andthescalarfieldcontributionTϑ ,which II. DYNAMICAL CHERN-SIMONS MODIFIED is provided by the following relationship GRAVITY

ϑ 1 ν Tµν = β (∇µϑ)(∇ν ϑ) − gµν (∇µϑ)(∇ ϑ) − gµν V (ϑ) . In this Section, we write down the field equations of the 2 Chern-Simons gravity, and present the Yunes-Pretorius (9) (YP) slowly-rotating solution found in [12]. Varying the action with respect to the scalar field ϑ, one obtains the equation of motion for the Chern-Simons coupling term, given by dV α A. Field equations of Chern-Simons theory β ∇ ∇µϑ = β − ∗RR. (10) µ dϑ 4 Consider the dynamical Chern-Simons modified grav- Note that the evolution of the CS coupling is not only ity theory provided by the action in the form governed by its stress-energy tensor, but also by the curvature of spacetime. In the nondynamical formulation of S = SEH + SCS + Sϑ + Smat . (1) CS modified gravity the constraint β =0isconsidered, 3

while in the dynamical framework, β is allowed to be ar- III. THERMAL EQUILIBRIUM RADIATION bitrary, so that Eq. (10) is now the evolution equation PROPERTIES OF THIN ACCRETION DISKS IN for the CS coupling field. STATIONARY AXISYMMETRIC SPACETIMES Considering the diffeomorphism invariance of the mat- µν ter part of the action, we have ∇µTmat =0,andtaking A. Stationary and axially symmetric spacetimes µν into account the Bianchi identities, i.e., ∇µG =0,provides the following conservation law The physical properties and the electromagnetic radi- 1 ation characteristics of particles moving in circular or- ∇ Cµν = − (∇ν ϑ) ∗RR. (11) bits around general relativistic bodies are determined by µ 8 the geometry of the spacetime around the compact object. For a stationary and axially symmetric geometry B. Rotating black hole solutions in Chern-Simons the metric is given in a general form by model 2 2 2 2 2 ds = gtt dt +2gtφ dtdφ + grr dr + gθθ dθ + gφφ dφ . In this paper, we consider the Yunes-Pretorius (YP) (14) slowly-rotating solution found in [12], where the CS In the equatorial approximation, which is the case of in- correction provides an effective reduction of the frame- terest for our analysis, the metric functions gtt, gtφ, grr, dragging around a black hole, in comparison with that of gθθ and gφφ only depend on the radial coordinate r,i.e., the Kerr solution. We will not analyze the Konno, Mat- |θ − π/2|1. suyama and Tanda (KMT) approximation [11], since the To compute the relevant physical quantities of thin ac- YP solution is taken to second order in the rotation pa- cretion disks, we determine first the radial dependence of the angular velocity Ω, of the specific energy E,andof rameter, and therefore gives more accurate results. e The Yunes-Pretorius (YP) approximation method em- the specific angular momentum L of particles moving in e ploys two schemes [12], namely, a small-coupling approxi- circular orbits in a stationary and axially symmetric ge- mation and a slow-rotation approximation. In particular, ometry through the geodesic equations. The latter take the small-coupling scheme treats the CS modification as a the following form [24] small deformation of general relativity. The slow-rotation scheme expands the background perturbations in powers dt Eg + Lg = e φφ e tφ , (15) of the Kerr rotation parameter a,andthebackground 2 dτ gtφ − gttgφφ metric is formalized via the Hartle-Thorne approxima- dφ Eg + Lg tion [13]. We refer the reader to [12] for details, and − tφ tt = e2 e , (16) present the final metric given by dτ gtφ − gttgφφ 2 2 2 2 dr E gφφ +2ELgtφ + L gtt 2 2M 2a M 2 2 g = −1+ e e e e . (17) ds = − 1 − + cos θ dt rr 2 − r r3 dτ gtφ gttgφφ 2M −1 " a2 2M −1!# + 1 − 1+ cos2 θ − 1 − dr2 From Eq. (17) one can introduce an effective potential r r2 r term as 2 4Ma 2 10 ξa 12M 27M 2 2 2 − sin θ − 1+ + sin θ dtdφ E gφφ +2ELgtφ + L gtt 4 2 − r 8 r 7r 10r Veff (r)= 1+ e 2 e e e . (18) gtφ − gttgφφ + r2 + a2 cos2 θ dθ2 2M For stable circular orbits in the equatorial plane + r2 sin2 θ + a2 sin2 θ 1+ sin2 θ dφ2 ,(12) r the following conditions must hold: Veff (r)=0and Veff, r(r)=0,wherethecommainthesubscriptde- with ξ = α2/(κβ). notes a derivative with respect to the radial coordinate r. In the following, we compare the properties of the met- These conditions provide the specific energy, the specific ric given by Eq. (12) with the standard general relativistic angular momentum and the angular velocity of particles Kerr metric, respectively, which in the equatorial approx- moving in circular orbits for the case of spinning general imation can be written as [22] relativistic compact spheres, given by

2 −1 2 2 2 −1 2 2 g + g Ω ds = −DA dt + r A (dφ − ωdt) + D dr + dz , E = − tt tφ , (19) 2 (13) e p−gtt − 2gtφΩ − gφφΩ where the coordinate z = r cos θ was used to replace θ, gtφ + gφφΩ 2 −4 2 −6 −2 2 −4 L = , (20) and A =1+a∗x +2a∗x and D =1− 2x + a∗x , 2 e p−gtt − 2gtφΩ − gφφΩ respectively. The dimensionless coordinate x is deﬁned 2 as x = pr/M,andthespinparametera∗ is deﬁned as dφ −gtφ,r + p(gtφ,r) − gtt,rgφφ,r 2 Ω= = . (21) a∗ = J/M = a/M. dt gφφ,r 4

The marginally stable orbit around the central object can black hole, is found to be 42% for rapidly rotating black be determined from the further condition Veff, rr(r)=0, holes, whereas the efficiency is 40% with photon capture which provides the following important relationship in the Kerr potential [23]. The accreting matter in the steady-state thin disk 0=(g2 − g g )V tφ tt φφ eff,rr model is supposed to be in thermodynamical equilib- = E2g +2ELg + L2g rium. Therefore the radiation emitted by the disk sur- e φφ,rr e e tφ,rr e tt,rr −(g2 − g g ) , (22) face can be considered as a perfect black body radiation, tφ tt φφ ,rr where the energy flux is given by F (r)=σT 4(r)(σ is 2 the Stefan-Boltzmann constant), and the observed lumi- where gtφ − gttgφφ (appearing as a cofactor in the metric determinant) never vanishes. By inserting Eqs. (19)-(21) nosity L (ν)hasaredshiftedblackbodyspectrum[28]: into Eq. (22) and solving this equation for r,weobtain the radii of the marginally stable orbits, once the metric 8 rf 2π ν3rdφdr L (ν)=4πd2I (ν)= cos γ Z Z e . coefficients gtt, gtφ and gφφ are explicitly given. − π ri 0 exp (νe/T ) 1 (25) Here d is the distance to the source, I(ν)isthethermal B. Physical properties of thin accretion disks energy flux radiated by the disk, γ is the disk inclination angle, and ri and rf indicate the position of the inner and For the thin accretion disk, it is assumed that its ver- outer edge of the disk, respectively. We take ri = rms and tical size is negligible, as compared to its horizontal ex- rf →∞, since we expect the flux over the disk surface tension, i.e, the disk height H,definedbythemaximum vanishes at r →∞for any kind of general relativistic half thickness of the disk, is always much smaller than compact object geometry. The emitted frequency is given the characteristic radius r of the disk, H r.Thethin by νe = ν(1 + z), and the redshift factor can be written disk has an inner edge at the marginally stable orbit of as the compact object potential, and the accreting plasma has a Keplerian motion in higher orbits. 1+Ωr sin φ sin γ 1+z = , (26) In steady state accretion disk models, the mass ac- 2 p−gtt − 2Ωgtφ − Ω gφφ cretion rate M˙ 0 is assumed to be a constant that does not change with time. The radiation flux F emitted by where we have neglected the light bending [35, 36]. the surface of the accretion disk can be derived from the conservation equations for the mass, energy and angular momentum, respectively. Then the radiant energy F (r) over the disk is expressed in terms of the specific energy, IV. ELECTROMAGNETIC SIGNATURES OF of the angular momentum, and of the angular velocity of ACCRETION DISKS AROUND the particles orbiting in the disk [20, 22], SLOWLY-ROTATING BLACK HOLES IN DYNAMICAL CHERN-SIMONS GRAVITY ˙ r M0 Ω,r Z F (r)=− √ (E − ΩL)L,rdr , (23) 4π −g (E − ΩL)2 r e e e e e ms Close to the equatorial plane of the slowly-rotating black holes, one can introduce the coordinate z = r cos θ where M˙ is the mass accretion rate, measuring the rate 0 describing “the height above the equatorial plane” and at which the rest mass of the particles flows inward write the metrics given by Eq. (12) in the form through the disk with respect to the coordinate time t and rms is the marginally stable orbit obtained from 2 2 Ma 1+h2(r) 2 Eq. (22). ds = −f(r)dt − 4 [1 + h1(r)]dtdφ + dr Another important characteristics of the mass accre- r f(r) 2 2 2 tion process is the efficiency with which the central object +r [1 + h3(r)]dφ + dz , (27) converts rest mass into outgoing radiation. This quantity is defined as the ratio of the rate of the radiation energy where f(r)=1− 2M/r is the Schwarzschild form factor, of photons, escaping from the disk surface to infinity, and and the rate at which mass-energy is transported to the central compact general relativistic object, both measured 5 ξ 12 M 27 M 2 h (r)=− 1+ + , (28) at infinity [20, 22]. If all the emitted photons can escape 1 16 Mr3 7 r 10 r2 to infinity, the efficiency is given in terms of the specific a2 energy measured at the marginally stable orbit rms, h (r)=− , (29) 2 r2f(r) =1− E . (24) 2 ems a 2M h3(r)= 1+ (30) For Schwarzschild black holes the efficiency is about r2 r 6%, whether the photon capture by the black hole is considered, or not. Ignoring the capture of radiation by the for the YP metric. 5

A. Constants of motion B. Flux and temperature distribution

If we insert the metric components of Eq. (27) into In Figs. 1 we present the flux distribution for the the expressions (19)-(21) of the specific energy, of the slowly-rotating Kerr black holes, and for the slowly specific angular momentum, and of the angular velocity, rotating YP solution, respectively. We consider the we obtain mass accretion driven by black holes with a total mass 6 of M =10M ,andwithamassaccretionrateof − f +2Mar 1Ω(1 + h ) ˙ −12 E = 1 ,(31) M0 =10 M /year. In units of the Eddington ac- −1 2 2 17 e pf +4Mar Ω(1 + h1) − r Ω (1 + h3) cretion rate M˙ Edd =1.5 × 10 (M/M )g/swehave −1 2 ˙ −10 ˙ −2Mar (1 + h )+r Ω(1 + h ) M0 =4.22 × 10 MEdd.Thespinparametera∗,runs L = 1 3 ,(32) −1 2 2 from 0.1 to 0.4, whereas the coupling constant of the CS e pf +4Mar Ω(1 + h1) − r Ω (1 + h3) gravity is set to ξ =28M 4,56M 4,112M 4 and 168M 4, Ma " s r3 # respectively. − ± Ω= 3 H1(r) H1(r)+ 2 , (33) 2r Ma H2(r) The plots show that the energy flux profiles of the disks in the CS modified gravity models deviate from the with slowly-rotating general relativistic Kerr black hole case. For the smallest values of ξ,theinneredgeoftheaccre- 1+h1 − rh1,r tion disk is located at somewhat higher radius than the H1(r)= , inner edge of the disk around the Kerr black hole (the H2(r) 1 location of the marginally stable orbits can be found in H (r)= (1 + h + rh ) . Table I, respectively, considered below). As the quanti- 2 2 3 3,r ties E, L and Ω in the flux integral (23) are still close for thee CSe model of gravity to those for the general rel- As Eqs. (31)-(33) show, the constants of motion for the ativistic case, for the lower boundary rms,GR

√ r 4M 2a2 C. Disk spectra and conversion efficiency −g = + r2. GR r2 − 2Mr In Fig. 3, we present the spectral energy distribution of As a result, the properties of particles orbiting the the disk radiation around the slowly-rotating black holes equatorial plane of slowly-rotating black holes in the for the general relativistic case, and for the CS modified standard general relativistic theory and in CS modified gravity. gravity are essentially the same. The only difference is The plots show that with the increase of the coupling in the location of the marginally stable orbits, which are constants of the CS gravity, the cut-off frequency of the strongly affected by the coupling. Since the inner edge of spectra decreases, from its value corresponding to the 14 athinaccretiondiskissupposedtobeattheradiusrms, Kerr black hole, to lower frequencies of the order of 10 the radial profile of the energy flux radiated over the disk Hz. Similarly to the case of the flux profiles, the effects surface can indicate the differences in the mass accretion of the CS coupling on the spectral cut-off are stronger for processes in the general relativistic theory, and in its CS black holes rotating faster than for very slowly-rotating type modification, respectively. black holes. For the radiation of the accretion disks 6

9 9 Kerr BH Kerr BH 8 ξ=28 M4 8 ξ=28 M4 ξ 4 ξ 4

] =56 M ] =56 M

2 7 4 2 7 4 - ξ=112 M - ξ=112 M m ξ 4 m ξ 4 c 6 =168 M c 6 =168 M 1 1 - - s 5 a =0.1 s 5 a =0.2 g * g * r r e e

7 4 7 4 0 0 1 1 [ 3 [ 3 ) ) r r ( (

F 2 F 2 1 1 0 0 5 6 7 8 10 20 40 5 6 7 8 10 20 40 r/M r/M

9 9 Kerr BH Kerr BH 8 ξ=28 M4 8 ξ=28 M4 ξ 4 ξ 4

] =56 M ] =56 M

2 7 4 2 7 4 - ξ=112 M - ξ=112 M m ξ 4 m ξ 4 c 6 =168 M c 6 =168 M 1 1 - - s 5 a =0.3 s 5 a =0.4 g * g * r r e e

7 4 7 4 0 0 1 1 [ 3 [ 3 ) ) r r ( (

F 2 F 2 1 1 0 0 5 6 7 8 10 20 40 5 6 7 8 10 20 40 r/M r/M

FIG. 1: The energy flux from accretion disks around slowly-rotating black holes for different spin parameters in the general relativity, and in the modified CS theory of gravity with the YP approximation. The coupling constant ξ is running from 28M 4 4 to 168M .Thespinparametersaresettoa∗ =0.1(upperlefthandplot),a∗ =0.2(upperrighthandplot),a∗ =0.3(lower 6 left hand plot) and a∗ =0.4(lowerrighthandplot),respectively.Thetotalblackholemassis10M and the mass accretion −12 −10 rate is 10 M /year=4.22 × 10 M˙ Edd. around black holes this means that the CS theory pro- the Kerr black holes provide a more efficient engine for duces rather similar disk spectra as in standard general the transformation of the energy of the accreting mass relativity, even that with increasing coupling constants into radiation than their slowly-rotating counterparts in the radial distributions of the fluxes differ to some ex- the modified CS theory of gravity, no matter what ap- tent for these two theories (see the top left hand plot in proximation is used. Figs. 1, and 3, respectively). In Table I we also present the conversion efficiency of the accreting mass into radiation for the case when V. DISCUSSIONS AND FINAL REMARKS the photon capture by the rotating black hole is ignored. 4 The variation of as a function of ξ/M is presented in In the present paper we have considered the basic phys- Fig. 4. ical properties of matter forming a thin accretion disk The value of measures the efficiency of energy gen- in slowly-rotating black hole spacetimes in the context erating mechanism by mass accretion. The amount of of the dynamical formulation of CS modified theories energy released by matter leaving the accretion disk of gravity. The physical parameters of the disk – en- and falling down the black hole is the binding energy ergy flux, temperature distribution and emission spec- E(r)|r=r of the black hole potential. trum profiles – have been explicitly obtained for several e ms Table I shows that is always higher for rotating gen- values of the coupling constant for the YP solution. Due eral relativistic black holes than for their counterparts in to the differences in the spacetime structure, the CS black CS modified gravity. As the Kerr black holes spin up, the holes present some very important differences with re- accreted mass-radiation conversion efficiency raises from spect to the disk properties, as compared to the standard about 6%, the characteristic value of the mass accretion general relativistic Kerr case. We have also shown that of the static black holes, to 7.5%. This feature is much the Kerr black holes also provide a more efficient engine more moderate for the rotating black holes in the CS the- for the transformation of the energy of the accreting mass ory: with increasing rotational velocity, also increases; into radiation than their slowly-rotating counterparts in however, the rate of this increase becomes smaller for the modified CS theory of gravity. stronger CS coupling. However, these values show that It is generally expected that most of the astrophysical 7

12 12 Kerr BH Kerr BH 11 ξ=28 M4 11 ξ=28 M4 ξ=56 M4 ξ=56 M4 10 ξ=112 M4 10 ξ=112 M4 ξ=168 M4 ξ=168 M4 ] ]

K 9 K 9

2 a =0.1 2 a =0.2

0 * 0 *

1 8 1 8 [ [ ) ) r r ( 7 ( 7 T T 6 6 5 5 4 4 5 6 7 8 10 20 40 5 6 7 8 10 20 40 r/M r/M

12 12 Kerr BH Kerr BH 11 ξ=28 M4 11 ξ=28 M4 ξ=56 M4 ξ=56 M4 10 ξ=112 M4 10 ξ=112 M4 ξ=168 M4 ξ=168 M4 ] ]

K 9 K 9

2 a =0.3 2 a =0.4

0 * 0 *

1 8 1 8 [ [ ) ) r r ( 7 ( 7 T T 6 6 5 5 4 4 5 6 7 8 10 20 40 5 6 7 8 10 20 40 r/M r/M

FIG. 2: The temperature distribution over the accretion disks around slowly-rotating black holes for diﬀerent spin parameters in general relativity and in the modiﬁed CS theory of gravity with the YP approximation. The values of M, a∗ and M˙ 0 are the same as in Fig. 1.

1034 1034

1033 1033 ] ] 1 1 - - s s g g r r

e 32 e 32 [ 10 [ 10 ) )

ν a*=0.4=0.1 ν a*=0.2 ( ( L L

ν Kerr BH ν Kerr BH 1031 ξ=28 M4 1031 ξ=28 M4 ξ=56 M4 ξ=56 M4 ξ=112 M4 ξ=112 M4 ξ=168 M4 ξ=168 M4 1030 1030 1013 1014 1013 1014 ν [Hz] ν [Hz]

1034 1034

1033 1033 ] ] 1 1 - - s s g g r r

e 32 e 32 [ 10 [ 10 ) )

ν a*=0.3 ν a*=0.4 ( ( L L

ν Kerr BH ν Kerr BH 1031 ξ=28 M4 1031 ξ=28 M4 ξ=56 M4 ξ=56 M4 ξ=112 M4 ξ=112 M4 ξ=168 M4 ξ=168 M4 1030 1030 1013 1014 1013 1014 ν [Hz] ν [Hz]

FIG. 3: The accretion disks spectra for slowly-rotating black holes with diﬀerent spin parameters in general relativity and in the modiﬁed CS theory of gravity with the YP approximation. The values of M, a∗ and M˙ 0 are the same as in Fig. 1. 8

7 a*=0.1 a =0.2 6.8 * a*=0.3 a =0.4 6.6 *

] 6.4 % [

ε 6.2

5.8

5.6 20 40 60 80 100 120 140 160 ξ / M4

FIG. 4: The conversion eﬃciency as a function of the CS coupling parameter ξ/M4 for diﬀerent values of the spin parameter.

4 a∗ ξ/M rin/M black holes comes from the very long baseline interfer- 0.1 - 5.6739 0.0606 ometry (VLBI) imaging of molecular H2Omasersinthe 28 5.7936 0.0599 active galaxy NGC 4258 [38]. This imaging, produced 56 5.8952 0.0592 by Doppler shift measurements assuming Keplerian motion of the masering source, has allowed a quite accurate 112 6.0762 0.0581 estimation of the central mass, which has been found to 168 6.2250 0.0571 7 be a 3.6 × 10 M super massive dark object, within 0.13 0.2 - 5.3315 0.0646 parsecs. Hence, important astrophysical information can 28 5.6122 0.0626 be obtained from the observation of the motion of the 56 5.8226 0.0611 gas streams in the gravitational field of compact objects. 112 6.1405 0.0588 The flux and the emission spectrum of the accretion 168 6.3807 0.0572 disks around compact objects satisfy some simple scaling 0.3 - 4.9818 0.0694 relations, with respect to the simple scaling transforma- 28 5.4662 0.0653 tion of the accretion rate and mass. In order to analyze 56 5.7744 0.0628 the scaling properties of the physical parameters of the accretion disks we introduce the scale invariant dimen- 112 6.1951 0.0595 sionless coordinate x = r/M.Thenthefunctionsh1(x) 168 6.5025 0.0573 and h3(x)givenbyEqs.(28)and(30)dependonlyonthe 0.4 - 4.6182 0.0751 scale invariant dimensionless spin parameter a∗ = a/M, 28 5.3546 0.0679 and they have no explicit dependence on the mass M.As 56 5.7456 0.0643 aconsequence,inEqs.(19)-(21),onlythespecificangular 112 6.2550 0.0601 momentum and the rotational frequency have an explicit −1 168 6.6100 0.0574 dependence on M,intheformL ∝ M and Ω ∝ M , e whereas E does depend only on a∗.Foranyrescaling e −1 M2 = αM1 of the mass of the star we obtain x2 = α x1, TABLE I: The inner edge of the accretion disk and the effi- and the relations ciency for slowly-rotating black holes in general relativity and in the CS modified theory of gravity with the YP approxima- ˜ ˜ tion. The lines where the value of ξ is not defined correspond E (M2; x2)=E(M1)(x1), L (M2; x2)=αL (M1; x1) , e e to the general relativistic case. (34) and

−1 objects grow substantially in mass via accretion. Re- Ω(M2; x2)=α Ω(M1; x1) , (35) cent observations suggest that around most of the active galactic nuclei (AGN’s) or black hole candidates there ˙ (2) ˙ (1) exist gas clouds surrounding the central far object, and respectively. For any rescaling M0 = βM0 of the ac- an associated accretion disk, on a variety of scales from cretion rate, these relations give atenthofaparsectoafewhundredparsecs[37].These β clouds are assumed to form a geometrically and opti- F M , M˙ (2) (x )= F M , M˙ (1) (x ) , (36) cally thick torus (or warped disk), which absorbs most 2 0 2 α2 1 0 1 of the ultraviolet radiation and the soft x-rays. The most powerful evidence for the existence of super massive as the scaling relation of the ﬂux integral given by 9

Eq. (23). Then the temperature scales like have an event horizon. Therefore the study of the accretion processes by compact objects is a powerful indicator β 1/4 of their physical nature. However, up to now, the obser- T M , M˙ (2) (x )= T M , M˙ (1) (x ) . 2 0 2 α2 1 0 1 vational results have confirmed the predictions of general (37) relativity mainly in a qualitative way. With the present observational precision one cannot distinguish between For the maximum of the luminosity L we have νmax ∝ T , 3 the different classes of compact/exotic objects that ap- which gives L (νmax) ∝ νmax.Asthefrequencyscales like the temperature, we obtain that pear in the theoretical framework of general relativity [29]. β 3/4 However, important technological developments may L M , M˙ (2) (ν )= L M , M˙ (1) (ν ) , 2 0 2 α2 1 0 1 allow one to image black holes and other compact ob- (38) jects directly [39]. In principle, detailed measurements with of the size and shape of the silhouette could yield information about the mass and spin of the central object, β 1/4 and provide invaluable information on the nature of the ν M , M˙ (2) = ν M , M˙ (1) . (39) 2 2 0 α2 1 1 0 accretion flows in low luminosity galactic nuclei. The spectrum in black hole systems can be dominated On the other hand, the flux is proportional to the ac- by the disc emission [40]. Recently, the RXTE satellite cretion rate M˙ 0,andthereforeanincreaseintheac- has provided a large number of data of the X-ray obser- cretion rate leads to a linear increase in the radiation vations of the accretion flows in galactic binary systems. emission flux from the disk. For a simultaneous scal- there is also a huge increase of the radio data for these ing of both the accretion rate M˙ 0 and of the mass of systems. The behavior of the spectrum in such systems the black hole M,themaximumofthefluxscalesas is consistent with the existence of a last stable orbit, and 2 F → M˙ 0/M F ,withalltheothercharacteristicsof such data can be used to estimate the black hole spin. At high luminosities these systems can also show very differ- the flux unchanged. Thus, if for a black hole mass with ent spectra [40]. Changes in the spectra of the disks are mass M =106M and by considering a mass accre- 1 driven by a changing geometry. Presently there exists an ˙ (1) −12 × −10 ˙ tion rate of M0 =10 M /year=4.22 10 MEdd, enormous amount of data from the X-ray binary systems 4 in the case a∗ =0.1andξ =28M ,themaximum which can be used to test this assumption. Therefore the (1) 7 −1 −2 of the flux is Fmax =4× 10 erg s cm .In study of accretion processes in Low Mass X-ray Binaries the case of a black hole with mass M2 =10M and with well constrained thermal spectra could also lead to ˙ (2) −4 with an accretion rate M0 =10 M /year=4.22 × the possibility of discriminating between the various ex- −2 10 M˙ Edd,themaximumpositionofthefluxis tensions of standard general relativity. (2) (2) (2) (1) (1) (1) Hence the study of the accretion processes by compact Fmax = hM˙ /M / M˙ /M i Fmax =4× 0 2 0 2 objects is a powerful indicator of their physical nature. 1025 erg s−1 cm−2.Thescalinglawofthetemperature √ Since the energy flux, the temperature distribution of the ˙ 1/4 is T → M0 / M T .Duetothetemperaturescal- disk, the spectrum of the emitted black body radiation, ing, the maximum value of the spectrum increases, but as well as the conversion efficiency show, in the case of the relative positions of the different spectral curves does the Chern-Simons theory vacuum solutions, significant not change. differences as compared to the general relativistic case, The determination of the accretion rate for an astro- the determination of these observational quantities could physical object can give a strong evidence for the exis- discriminate, at least in principle, between standard gen- tence of a surface of the object. A model in which Sgr eral relativity and Chern-Simons gravity, and constrain 6 A*, the 3.7 × 10 M super massive black hole candidate the parameters of the model. at the Galactic center, may be a compact object with a thermally emitting surface was considered in [39]. Given the very low quiescent luminosity of Sgr A* in the near- infrared, the existence of a hard surface, even in the limit Acknowledgments in which the radius approaches the horizon, places a se- vere constraint on the steady mass accretion rate onto the We would like to thank the two anonymous referees −12 −1 source, M˙ ≤ 10 M yr .Thislimitiswellbelowthe whose comments and suggestions helped us to signifi- minimum accretion rate needed to power the observed cantly improve the manuscript. The work of T. H. was −10 submillimeter luminosity of Sgr A*, M˙ ≥ 10 M yr. supported by the General Research Fund grant number Thus, from the determination of the accretion rate it fol- HKU 701808P of the government of the Hong Kong Spe- lows that Sgr A* does not have a surface, that is, it must cial Administrative Region. 10

[1] E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. folini, E. C. Pavlis, Nature 431 (2004) 958. Phys. D 15,1753(2006);T.P.SotiriouandV.Faraoni, [17] K. Konno, T. Matsuyama, Y. Asano and S. Tanda, Phys. arXiv:0805.1726 [gr-qc]; S. Nojiri and S. D. Odintsov, Int. Rev. D 78,024037(2008). J. Geom. Meth. Mod. Phys. 4,115(2007);K.Koyama, [18] N. Yunes and D. N. Spergel, Phys. Rev. D 80,042004 Gen. Rel. Grav. 40,421(2008);F.S.N.Lobo, (2009). arXiv:0807.1640 [gr-qc]. [19] V. Cardoso and L. Gualtieri, Phys. Rev. D 80,064008 [2] S. Capozziello, V. F. Cardone and A. Troisi, JCAP (2009). 0608,001(2006);S.Capozziello,V.F.Cardoneand [20] I. D. Novikov and K. S. Thorne, in Black Holes, ed. C. A. Troisi, Mon. Not. R. Astron. Soc. 375,1423(2007); DeWitt and B. DeWitt, New York: Gordon and Breach A. Borowiec, W. Godlowski and M. Szydlowski, Int. J. (1973). Geom. Meth. Mod. Phys. 4 (2007) 183; C. F. Martins [21] N. I. Shakura and R. A. Sunyaev, Astron. Astrophys. 24, and P. Salucci, Mon. Not. Roy. Astron. Soc. 381,1103 33 (1973). (2007). C. G. Boehmer, T. Harko and F. S. N. Lobo, As- [22] D. N. Page and K. S. Thorne, Astrophys. J. 191,499 tropart. Phys. 29,386(2008).C.G.Boehmer,T.Harko (1974). and F. S. N. Lobo, JCAP 0803,024(2008). [23] K. S. Thorne, Astrophys. J. 191,507(1974). [3] A. Lue, L. M. Wang and M. Kamionkowski, Phys. Rev. [24] T. Harko, Z. Kovacs and F. S. N. Lobo, Phys. Rev. D 78, Lett. 83,1506(1999). 084005 (2008); T. Harko, Z. Kovacs and F. S. N. Lobo, [4] Phys. Rev. D 68,104012(2003). Phys. Rev. D 79,064001(2009). [5] S. Alexander and N. Yunes, Phys. Rept. 480,1(2009). [25] S. Bhattacharyya, A. V. Thampan and I. Bombaci, As- [6] S. H. S. Alexander and S. J. J. Gates, JCAP 0606,018 tron. Astrophys. 372,925(2001). (2006); B. A. Campbell, N. Kaloper, R. Madden and [26] Z. Kovacs, K. S. Cheng and T. Harko, Astron. Astrophys. K. A. Olive, Nucl. Phys. B 399,137(1993). 500,621(2009). [7] S. S. Alexander, M. E. Peskin and M. M. Sheikh-Jabbari, [27] Z. Kovacs, K. S. Cheng and T. Harko, arXiv:0908.2672, Phys. Rev. Lett. 96,081301(2006);S.Alexanderand to appear in MNRAS (2009). N. Yunes, Phys. Rev. D 77,124040(2008);M.B.Cantch- [28] D. Torres, Nucl. Phys. B 626,377(2002). eﬀ, Phys. Rev. D 78,025002(2008);S.Alexander, [29] Y. F. Yuan, R. Narayan and M. J. Rees, Astrophys. J. L. S. Finn and N. Yunes, Phys. Rev. D 78,066005(2008); 606,1112(2004). B. Tekin, Phys. Rev. D 77,024005(2008);D.Guar- [30] F. S. Guzman, Phys. Rev. D 73,021501(2006). rera and A. J. Hariton, Phys. Rev. D 76,044011(2007); [31] T. Harko, Z. Kovacs and F. S. N. Lobo, Class. Quant. D. Grumiller and N. Yunes, Phys. Rev. D 77,044015 Grav. 26,215006(2009). (2008); N. Yunes and C. F. Sopuerta, Phys. Rev. D 77, [32] C. S. J. Pun, Z. Kovacs and T. Harko, Phys. Rev. D 78, 064007 (2008). 084015 (2008). [8] S. Alexander and N. Yunes, Phys. Rev. D 75,124022 [33] C. S. J. Pun, Z. Kovacs and T. Harko, Phys. Rev. D 78, (2007). 024043 (2008). [9] S. Alexander and N. Yunes, Phys. Rev. Lett. 99,241101 [34] T. Harko, Z. Kovacs and F. S. N. Lobo, Phys. Rev. D (2007). 80,044021(2009). [10] K. Konno, T. Matsuyama, and S. Tanda, Phys. Rev. D [35] J. P. Luminet, Astron. Astrophys. 75,228(1979). 76,024009(2007). [36] S. Bhattacharyya, R. Misra, and A. V. Thampan, Astro- [11] K. Konno, T. Matsuyama and S. Tanda, Prog. Theor. phys. J. 550,841(2001). Phys. 122,561(2009). [37] C. M. Urry and P. Padovani, Publ. Astron. Soc. of the [12] N. Yunes and F. Pretorius, Phys. Rev. D 79,084043 Paciﬁc 107,803(1995). (2009). [38] M. Miyoshi, J. Moran, J. Herrnstein, L. Greenhill, N. [13] J. B. Hartle and K. S. Thorne, Astrophys. J. 153,807 Nakai, P. Diamond and M. Inoue, Nature 373,127 (1968); K. S. Thorne and J. B. Hartle, Phys. Rev. D 31, (1995). 1815 (1984). [39] A. E. Broderick and R. Narayan, Astrophys. JL. 636, [14] S. Alexander and N. Yunes, Phys. Rev. D 75,124022 L109 (2006). (2007). [40] C. Done, M. Gierlinski, and A. Kubota, Astron. Astro- [15] T. L. Smith, A. L. Erickcek, R. R. Caldwell and phys. Rev. 15,1(2007). M. Kamionkowski, Phys. Rev. D 77,024015(2008). [16] I. Ciufolini, Gravitomagnetism, arXiv:0704.3338; I. Ciu-