A&A 551, A4 (2013) Astronomy DOI: 10.1051/0004-6361/201220378 & c ESO 2013 Astrophysics Accretion disks around black holes in modified strong gravity D. Pérez1,G.E.Romero1,2, and S. E. Perez Bergliaffa3 1 Instituto Argentino de Radioastronomía,Camino Gral Belgrano Km 40 C.C.5, 1984 Villa Elisa, Buenos Aires, Argentina e-mail: [email protected] 2 Facultad de Ciencias Astronómicas y Geofísicas, UNLP, Paseo del Bosque s/n CP, 1900 La Plata, Buenos Aires, Argentina e-mail: [email protected] 3 Departamento de Física Teórica, Instituto de Física, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, Maracanã, Rio de Janeiro, CEP: 20550-900, Brasil e-mail: [email protected] Received 13 September 2012 / Accepted 18 November 2012 ABSTRACT Context. Stellar-mass black holes offer what is perhaps the best scenario to test theories of gravity in the strong-field regime. In particular, f (R) theories, which have been widely discus in a cosmological context, can be constrained through realistic astrophysical models of phenomena around black holes. Aims. We aim at building radiative models of thin accretion disks for both Schwarzschild and Kerr black holes in f (R) gravity. Methods. We study particle motion in f (R)-Schwarzschild and Kerr space-times. Results. We present the spectral energy distribution of the accretion disk around constant Ricci scalar f (R) black holes, and constrain specific f (R) prescriptions using features of these systems. Conclusions. A precise determination of both the spin and accretion rate onto black holes along with X-ray observations of their thermal spectrum might allow to identify deviations of gravity from general relativity. We use recent data on the high-mass X-ray binary Cygnus X-1 to restrict the values of the parameters of a class of f (R) models. Key words. black hole physics – accretion, accretion disks – gravitation 1. Introduction Although the present revival of f (R) theories is mainly due to their use in the description of phenomena that take place for General relativity (GR) is consistent, in some cases with good low values of the Riemann curvature, these theories have also precision, with observational results (see for instance Will 2006). been applied to gravity in the opposite regime. As there is no However, an energy momentum tensor representing exotic mat- direct evidence of the behaviour of the gravitational field for ter (loosely called “dark energy”, e.g. Li et al. 2011)mustbe very large values of the curvature, the early universe and com- introduced in the right hand side of Einstein’s equations to fit pact objects offer the possibility to find deviations from GR. the currently available data when these are interpreted in the Among the studies in modified gravity in the strong regime, framework of the standard cosmological model (based on GR). we can mention the successful inflationary model based on the Dark energy can be modelled by a cosmological constant, or by R + αR2 theory (Starobinsky 1980), and the related studies of re- a scalar field with an equation of state given by p = ωDEρ,where heating (Motohashi & Nishizawa 2012) and particle production ωDE < −1/3 (Biswas et al. 2010a,b). None of these descriptions (Arbuzova et al. 2012) in the early universe. Also of importance is free of problems, since the energy density associated with the is the treatment of neutron stars (Cooney et al. 2010) and black cosmological constant that is inferred from astronomical obser- hole solutions. vations is approximately 120 orders of magnitude lower than the Different aspects of black hole physics in f (R) theo- value predicted by field theory (e.g. Weinberg 1989; Capozziello ries have been discussed in the literature by Psaltis et al. & Faraoni 2010), whereas the scalar field has features that are at (2008a), Hendi & Momeni (2011), Myung (2011), Myung et al. odds with the scalar fields of particle physics (Sotiriou & Faraoni (2011), Moon et al. (2011a,b), Hendi et al. (2012), and Habib 2010). Mazharimousavi et al. (2012). Static and spherically symmet- Adifferent approach to explain the cosmological data is to ric black hole solutions were obtained via perturbation theory modify the field equations of the gravitational field, in such a by de la Cruz-Dombriz et al. (2009), whereas black holes with way that the ensuing theory differs from Einstein’s in the low- these symmetries have been studied by means of a near-horizon curvature regime. Since there is no a priori fundamental reason analysis by Perez Bergliaffa & De Oliveira (2011). Finally, to restrict the gravitational Lagrangian to a linear function of the f (R)-Kerr-Newman black holes solutions with constant Ricci Ricci scalar R (see for instance Magnano et al. 1987), more gen- scalar have been recently studied by Cembranos and collabo- eral theories can be formulated using nonlinear functions of this rators (Cembranos et al. 2011). scalar. The so-called f (R) theories (e.g. Capozziello & Faraoni From an astrophysical point of view, thin accretion for the 2010) were first used to mimic the inferred accelerated expan- Schwarzschild space-time in f (R)-gravity has been discussed by sion of the universe by Capoziello (2002). Currently there is in Pun et al. (2008), without including the expected spectra of con- the literature a handful of f (R)-models in agreement with avail- crete astrophysical black holes or any comparison with observa- able data (De Felice & Tsujikawa 2010). tional data. Article published by EDP Sciences A4, page 1 of 15 A&A 551, A4 (2013) In the present work we investigate the existence of stable where: circular orbits in Schwarzschild and Kerr f (R) space-times with f (R ) constant Ricci scalar and analyze the main features of accretion Λ ≡ 0 − , (6) disks around these black holes. In particular, we present temper- f (R0) 1 ature and spectral energy distributions for Page-Thorne disks, and, by Eq.(5): and compare the results with those obtained using the standard Shakura-Sunyaev model. = 2 f (R0) · R0 (7) The paper is organized as follows. In Sect. 2 we provide a f (R0) − 1 brief review of f (R) theories of gravity. Circular orbits in both Schwarzschild and Kerr f (R) space-times with constant Ricci Hence, in this case, any f (R) theory is formally, but not phys- ically, equivalent to GR with a cosmological constant given by scalar are studied in Sect. 3. Section 4 is devoted to the calcu- 1 lation of the properties of accretion disks in these space-times. Eq. (6) . Consequences for some specific prescriptions for the function f We shall investigate the existence of stable circular orbits in are discussed in Sect. 5. We close with some considerations on Schwarzschild and Kerr f (R) space-times with constant Ricci the potential of astronomical observations to test the strong-field scalar in the next section. regime of gravity. 3. Circular orbits around a black hole in f(R) 2. f(R)gravity 3.1. f(R)-Schwarzschild space-time In f (R) gravity, the Lagrangian of the Hilbert-Einstein action, The Schwarzschild space-time metric in f (R) theories with con- given by: stant Ricci scalar R0 takes the form (Cembranos et al. 2011): 3 2 c √ 2GM c R0 S [g] = R −g d4 x, (1) ds2 = − c2 − − r2 (8) 16πG r 12 dr2 is generalized to: dt2 + + r2 dθ2 + sin2 θdφ2 , − 2MG − R0 2 1 c2r 12 r c3 √ S [g] = (R + f (R)) −g d4 x, (2) 16πG where R0,givenbyEq.(7), can take, in principle, any real value. Since we are looking for space-time metrics that may represent where g is the determinant of the metric tensor, and f (R)isan astrophysical black holes, we shall select those values of R0 that arbitrary function of the Ricci scalar. In the metric formalism the lead to acceptable solutions. field equations are obtained varying Eq. (2) with respect to the The radius r0 of the horizon follows from the condition metric: g00(r0) = 0. From Eq. (8), the values of r corresponding to the event horizon satisfy: + − 1 + Rμν(1 f (R)) gμν(R f (R)) (3) 2 3 2 2 c R0r − 12c r + 24GM = 0. (9) 16πG + (∇ ∇ − g ) f (R) + T = 0, In terms of the following adimensional quantities: μ ν μν c4 μν r ≡∇∇β = x ≡ , (10) where Rμν is the Ricci Tensor, β , f (R) d f (R)/dR,and r the energy momentum tensor is defined by: g ≡ 2 √ R0 R0rg, (11) −2 δ( −gLm) T = √ · (4) = 2 μν −g δgμν where rg GM/c , this equation takes the form: 3 R0x − 12x + 24 = 0. (12) Here, Lm stands for the matter Lagrangian. Equations (3) are a system of fourth-order nonlinear equa- We show in Fig. 1 the Ricci scalar as a function of the radial tions for the metric tensor gμν. An important difference between coordinate of the event horizon. We see that for R0 ∈ (0, 4/9) them and the Einstein field equations is that in f (R) theories the there is an inner black hole event horizon and an outer cosmo- Ricci scalar R and the trace T of the energy momentum tensor logical horizon, whereas for R ≤ 0 there is only one black ff 0 are di erentially linked, as can be seen by taking the trace of hole event horizon. The event and cosmological horizons col- Eq. (3), which yields: lapse for R0 = 4/9 and for larger values of the Ricci scalar 16πG naked singularities occur and hence, no black holes are possi- R(1 + f (R)) − 2(R + f (R)) − 3 f (R) + T = 0.