# Does black hole continuum spectrum signal higher curvature gravity in higher dimensions?

###### Abstract

Extra dimensions, which led to the foundation and inception of string theory, provide an elegant approach to force-unification. With bulk curvature as high as the Planck scale, higher curvature terms, namely gravity seems to be a natural addendum in the bulk action. These can not only pass the classic tests of general relativity but also serve as potential alternatives to dark matter and dark energy. With interesting implications in inflationary cosmology, gravitational waves and particle phenomenology it is worth exploring the impact of extra dimensions and higher curvature in black hole accretion. Various classes of black hole solutions have been derived which bear non-trivial imprints of these ultraviolet corrections to general relativity. This in turn gets engraved in the continuum spectrum emitted by the accretion disk around black holes. Since the near horizon regime of supermassive black holes manifest maximum curvature effects, we compare the theoretical estimates of disk luminosity with quasar optical data to discern the effect of the modified background on the spectrum. In particular, we explore a certain class of black hole solution bearing a striking resemblance with the well-known Reissner-Nordström de Sitter/anti-de Sitter/flat spacetime which unlike general relativity can also accommodate a negative charge parameter. By computing error estimators like chi-square, Nash-Sutcliffe efficiency, index of agreement, etc. we infer that optical observations of quasars favor a negative charge parameter which can be a possible indicator of extra dimensions. The analysis also supports an aymptotically de Sitter spacetime with an estimate of the magnitude of the cosmological constant whose origin is solely attributed to higher curvature terms in higher dimensions.

## 1 Introduction

General relativity (GR) is a classic example of a scientific theory that is elegant, simple and powerful. Till date, it is the most successful theory of gravity in explaining a plethora of observations namely, the perihelion precession of mercury, the bending of light, the gravitational redshift of radiation from distant stars, to name a few [1, 2, 3]. Very recently, the shadow of the black hole in M87 observed by the Event Horizon Telescope has further added to its phenomenal success [4, 5, 6]. Yet it is instructive to subject GR to further tests since it is marred with unresolved issues like singularities [7, 8, 9] and falls short in explaining the nature of dark energy and dark matter [10, 11, 12, 13, 14]. Moreover, the quantum nature of gravity is still elusive and ill-understood [15, 16, 17]. All this makes the quest for a more complete theory of gravity increasingly compelling such that it yields GR in the low energy limit. Consequently a surfeit of alternate gravity models are proposed which can potentially fulfill the deficiencies in GR. A viable alternate gravity theory must be free from ghost modes, be consistent with solar system based tests, should not engender a fifth force in local physics and should successfully explain observations that GR fails to address. The alternate gravity models which fulfill these benchmark can be broadly classified into three categories: (i) Modified gravity models where the gravity action is supplemented with higher curvature terms, e.g., f(R) gravity [18, 19, 20, 21], Lanczos-Lovelock models etc. [22, 23, 24, 25, 26] (ii) Extra-dimensional models that alter the effective 4-dimensional gravitational field equations due to the bulk Weyl stresses and higher order corrections to the stress-tensor [27, 28, 29, 30, 31, 32, 33] and (iii) Scalar-tensor theories of gravity which include the Brans-Dicke theory and the more general Horndeski models [34, 35, 36, 37].

In this work we will consider modifications to the gravity sector by introducing gravity in five dimensions. Among the various modified gravity models, theories have attracted the attention of physicists for a long time [18, 38, 39, 40] since they invoke the simplest modification to the Einstein-Hilbert action and yet exhibit sufficient potential to address a host of cosmological and astrophysical observations. These include, but are not limited to, the late time acceleration [41, 42] and the initial power-law inflation of the universe [43], the four cosmological phases [19, 44], the rotation curves of spiral galaxies [45, 46] and the detection of gravitational waves [47, 48, 49]. Although these models are plagued with ghost modes, certain models e.g. theory on a constant curvature hypersurface can be shown to be ghost free [50, 51, 52]. In addition, they can successfully surpass the solar system tests which only impose constraints on and hence on the model parameters [53, 54, 55].

Extra-dimensions on the other hand were mainly invoked to provide a framework to unify gravity and electromagnetism [56, 57, 58]. This subsequently provided a framework for string theory and M-theory that succeeded in unifying all the known forces under a single umbrella [59, 60, 61]. The large radiative corrections to the Higgs mass arising due to the huge disparity between the electro-weak scale and the Planck scale [62, 63, 64, 65, 66] led to the emergence of a diversity of string inspired brane-world models. Most of these models assume that the observable universe is confined in a 3-brane where all the Standard Model particles and fields reside while gravity permeates to the bulk [62, 64, 63, 67, 68, 65, 66]. They possess interesting phenomenological implications [69, 70, 71, 72, 73, 74] and distinct observational signatures including production of mini-black holes which can be tested in present and future collider experiments [75, 76]. In the galactic scale, they offer an alternative to the elusive dark matter [77, 45, 78, 79, 80] while in cosmology they have interesting implications in the inflationary epoch [81, 82, 83, 84, 85, 86, 87] and also serve as a possible proxy to dark-energy [88, 89, 90, 91, 31, 92, 93]. Since the ultraviolet nature of gravity is unknown, it is often believed that in the high energy regime, the deviations from Einstein gravity may manifest through the existence of extra dimensions. Moreover, the bulk curvature is expected to be as high as the Planck scale and hence higher order corrections to the gravity action should become relevant in the high energy regime.

In this work we consider a single braneworld scenario with a positive tension which is embedded in a five dimensional bulk containing gravity. The addition of higher curvature terms in higher dimensions cause substantial modification to the effective gravitational field equations on the brane [94, 30, 95, 27, 33] which are obtained from Gauss-Codazzi equation and the junction conditions [96]. Such deviations from Einstein’s equations are expected to become more conspicuous in the high energy/high curvature domain. Therefore, the near horizon regime of black holes where the curvature effects are maximum, seem to be an ideal astrophysical laboratory to test these models against observations.

Various classes of vacuum solutions of these field equations have been obtained [28, 29, 32, 97, 98] which possess distinct signatures of extra dimensions and gravity. In the event the vacuum solutions are static and spherically symmetric, the electric part of the Weyl tensor can be decomposed into terms involving “dark radiation” and “dark pressure”. Suitable integrability conditions lead to different classes of vacuum solutions which determine the spacetime geometry. The solutions thus derived exhibit substantial modification from the well-known Schwarzschild spacetime which are attributed to the non-local effects of the bulk Weyl tensor and the higher curvature terms in the action. These deviations in the background spacetime are sculpted in the continuum spectrum emitted from the accretion disk around black holes. In particular, since the curvature effects are maximum in supermassive black holes, the quasar continuum spectra can act as potential astrophysical probes to establish/falsify/constrain these models.

In a recent work [99] we explored an exact black hole solution in the brane with bulk Einstein gravity. It resembles the well-known Reissner-Nordström spacetime in general relativitywhere the tidal charge parameter can assume both signatures. By comparing the disk luminosity of quasars in such a background with the corresponding observations we conclude that a negative charge parameter is favored which is characteristic to braneworld black holes. Adding gravity in the bulk action adds a vaccum energy term to the aforesaid black hole solution where the cosmological constant owes its origin to terms involving higher curvature and higher dimensions. In this work we investigate the effect of such a spacetime on the quasar continuum spectrum which enables us to explore the signature of the tidal charge parameter in the presence of the cosmological constant term in the metric. Subsequently, we also derive constraints on the magnitude of the cosmological constant from quasar optical data. Further, we also investigate the effect of other black hole solutions on the quasar continuum spectrum, which are derived by altering the relations connecting the “dark radiation” and “dark pressure”.

The paper is organized as follows: In Section 2 we discuss the modifications induced in the gravitational field equations due to the presence of bulk gravity. The static, spherically symmetric, vaccum solutions of these field equations are reviewed in Section 3. In Section 4 we examine the properties of the black hole continuum spectrum in presence of the background spacetimes discussed in Section 3. Section 5 is dedicated to numerical analysis where the theoretically computed luminosities from the accretion disk of eighty quasars are compared with the corresponding observed values. Finally, we conclude with a summary and the discussion of our results in Section 6.

Notations and Conventions: Throughout this paper, the Greek indices denote the four dimensional spacetime and capitalized latin alphabets represent the five dimensional bulk indices. We will work in geometrized unit with and the metric convention will be mostly positive.

## 2 Static, spherically symmetric black hole solutions in higher dimensional gravity

In this section we consider gravity in the bulk action and derive the effective gravitational field equations on the brane. The bulk action assumes the form,

(1) |

where is the bulk metric, is the bulk Ricci scalar and is the matter Lagrangian. The bulk indices are denoted by capitalized latin alphabets e.g. A, B which run over all space-time dimensions while Greek letters denote the brane coordinates. The gravitational field equation obtained by varying the bulk action with respect to is given by,

(2) |

where is the bulk Ricci tensor, is the five dimensional gravitational constant and prime denotes derivative with respect to . The bulk energy-momentum tensor can be written as,

(3) |

where , the negative vacuum energy density on the bulk, the brane tension and the brane energy-momentum tensor are the sources of the gravitational field on the bulk. The various physical quantities on the bulk are projected onto the brane with the help of the projector . The brane is located at (where represents the extra coordinate) and the induced metric on the hypersurface is represented by .

In order to obtain the effective gravitational field equations on the brane, Gauss-Codazzi equation is used which connects the bulk Riemann tensor to that of the brane with the help of the projector and the extrinsic curvature tensor . The extrinsic curvature is related to the covariant derivative of the normalized normals to the brane and encodes the embedding of the brane into the bulk. The presence of a brane energy momentum tensor leads to a discontinuity in across the brane. Israel junction conditions and a orbifold symmetry relates this discontinuity in the extrinsic curvature to the brane energy momentum tensor. For a detailed derivation one is referred to [96, 29, 32, 33].

With the above considerations the effective four-dimensional gravitational field equations on the brane assume the form,

(4) |

where

(5) | ||||

(6) | ||||

(7) | ||||

(8) | ||||

(9) | ||||

(10) |

In Eq. (, and refer to the Ricci tensor and Ricci scalar on the brane while and represent the 4-dimensional cosmological constant and gravitational constant respectively. Eq. ( serves as the fine balancing relation of the Randall-Sundrum single brane model [67, 27] which enables the brane tension to be tuned appropriately with the bulk cosmological constant to yield de-Sitter, anti de-Sitter or flat branes. In Eq. (, represents higher order terms associated with the brane energy momentum tensor due to the local effects of the bulk on the brane. The term arises because of the presence of higher curvature terms in the bulk action. In the event , and we recover the projected field equations on the brane due to pure Einstein gravity in the bulk. The expression for can be simplified further by assuming that when the second term in Eq. ( vanishes (see for example [32]) such that,

(12) |

Since the bulk Ricci scalar is expected to be a well-behaved quantity, it can be expanded in a Taylor series around , i.e.,

(13) |

where the coefficients are constants since is independent of the brane coordinates. This implies that the derivatives of evaluated at in Eq. ( will result in a constant contribution independent of the brane coordinates.

The last term on the right hand side of Eq. ( is which epitomizes the electric part of the bulk Weyl tensor with its origin in the nonlocal effect from the free bulk gravitational field. It is the transmitted projection of the bulk Weyl tensor on the brane, such that with the property, . The conservation of matter energy-momentum tensor on the brane i.e , (where represents the brane covariant derivative) leads to the constraint , since as the bulk Ricci scalar depends only on .

The symmetry properties of allows an irreducible decomposition of the tensor in terms of a given 4-velocity field [100, 29],

(14) |

where with and is the projector orthogonal to . Note that , such that we retrieve general relativity in the limit [29]. In Eq. ( the scalar is often known as the “Dark Radiation” term. The second term on the right hand side of Eq. ( consists of a spatial vector whereas the third term consists of a spatial, tracefree, symmetric tensor .

In order to obtain vacuum solutions on the brane, the brane should be source free such that . Thus, the gravitational field equations on the brane reduce to,

(15) |

In such a scenario, the effective four-dimensional cosmological constant is given by while the conservation of energy-momentum tensor on the brane simplifies to, . Additionally, if the solutions are static, the term in Eq. ( should vanish such that the conservation of brane energy-momentum tensor leads to,

(16) |

where is the -acceleration and denotes covariant derivative on the space-like hypersurface orthonormal to . Further, if the solutions are spherically symmetric, we may write , while the term can be written as,

(17) |

where and (also known as the “Dark Pressure”) are scalar functions of the radial coordinate and is the unit radial vector.

In order to derive static, spherically symmetric solutions of Eq. ( we consider a metric ansatz of the form,

(18) |

and solve for , , and since Eq. ( satisfies Eq. ( and Eq. (. One can show that the solution of these equations lead to the following form for [32],

(19) |

where is an arbitrary integration constant and is defined as,

(20) |

From the form of it can be inferred that is the gravitational mass originating from the dark radiation and can be interpreted as the “dark mass” term. It is important to emphasize that in the limit , and , we get back the standard Schwarzschild solution and the constant of integration can then be identified with , where is the mass of the gravitating body.

Further, one can show that for a static, spherically symmetric spacetime the ordinary differential equations for dark radiation and dark pressure satisfy [32],

(21) |

and

(22) |

where and . Eq. ( and Eq. ( can be recast into a more convenient form namely,

(23) | ||||

(24) |

by defining the variables,

(25) |

Eq. ( and Eq. ( can be referred to as the differential equations governing the source terms on the brane. For a detailed derivation of the differential equations for the metric components and the source terms one is referred to [32, 29]. In the next section we shall review various static, spherically symmetric and vacuum solutions of Eq. ( on the brane.

## 3 Various classes of solutions on the brane

The source equations Eq. ( and Eq. ( for dark radiation and dark pressure cannot be solved simultaneously until we impose some further conditions on them. Hence, we choose some specific relations between dark radiation and dark pressure , necessarily defining the various equations of state in the framework of the brane world model. We will note that the different choices of equations of state will lead to very distinct solutions.

### 3.1 Case A: P = 0

This is the vanishing dark pressure case. The dark radiation and the dark mass can be evaluated by solving the coupled equations Eq. ( and Eq. (. With , these two equations simplify to,

(26) | ||||

(27) |

respectively. The above two equations can be combined to produce a single differential equation given by,

(28) |

Since is not a constant in Eq. ( we apply some approximate methods to find a solution for . By taking Laplace transformation of Eq. ( and using the convolution theorem we get an integral solution for ,

(29) |

with the associated functions,

(30) | |||

(31) | |||

(32) | |||

(33) |

where is an arbitrary point which can be associated with the vacuum boundary of a compact astrophysical object [32, 29] and .

Eq. ( can be solved by applying successive approximation methods. The zeroth order solution denoted by is derived by considering only the linear part of Eq. (. The full solution can thus be expressed as lim, ( being the order of the equation) such that the iterative solution at order is connected to the order by the following differential equation [32, 29],

(34) |

Once we determine the solution for we can derive the solution for the metric components by using the gravitational field equations on the brane and the condition for conservation of energy-momentum tensor. In the zeroth order, the static and spherically symmetric solution to the field equations is given by [32, 29],

(35) | ||||

(36) | ||||

(37) |

where is an arbitrary constant of integration. Since is positive Eq. ( implies that and consequently should be positive. Also, the component of the metric should be positive, which implies .

Iterating once more, we get the approximate expressions for and upto first order,

(38) | |||

(39) |

Since we are interested in the distances much smaller compared to the cosmological horizon , it is reasonable to assume . Under this assumption Eq. ( simplifies considerably,

(40) |

where . It is evident from Eq. ( that should be positive while can assume both signatures. Further, if we can perform a binomial expansion of Eq. ( giving rise to a solution of the form,

(41) |

where and . Note that the dependence on gravity comes from the parameter . Eq. ( can be rescaled such that the component of the metric assumes the form,

(42) |

where

(43) |

Therefore it is clear from Eq. ( that in the regime , the component of the approximate metric is very similar to the Schwarzschild spacetime in general relativity, although the ADM mass has contributions from the inertial mass as well as the higher curvature and higher dimension terms. The component solely determines the photon sphere and the radius of the marginally stable circular orbit of massive test particles. The photon sphere is obtained from the solution of,

(44) |

while the marginally stable circular orbit is evaluated from the solution of

(45) |

Note that should be positive, otherwise and becomes negative, which is unphysical. Since and are both positive, together they ensure that .

In order to simplify our calculations we scale the radial distance in units of the gravitational radius , such that Eq. ( assumes the form,

(46) |

where (with ). The deviation of the approximate metric from the Schwarzschild spacetime is manifested in the term, where

(47) |

where,

Since we are interested in black hole solutions the curvature singularity at must be covered by an event horizon. The radius of the event horizon is obtained from the real positive solutions of . Since is a fifth order algebraic equation, it always has at least one real root. For the real root to be positive we need to choose the values of , , and judiciously. From the previous discussion it is evident that and are always positive while and can assume any signature. Further constraints on the values of , , , and are established from the fact that .

The disadvantage of this choice of equation of state is that the metric does not represent an exact black hole solution. Following the same iterative procedure we can approximate the metric to second and the next higher orders. However, we have to work out the properties of this metric (namely the , and ) order by order which is not a desirable feature. In the next section we consider another choice of equation of state which will turn out to be more useful.

### 3.2 Case B: 2U + P = 0

In this section we consider an interesting scenario where the dark radiation, “” and the dark pressure “” satisfy the constraint . For this specific choice, Eq. ( leads to,

(48) |

Therefore the general solution for the dark pressure and the dark radiation is given by,

(49) |

where is an arbitrary constant of integration. Consequently, from Eq. ( the dark mass assumes the form,

(50) |

with the integration constant . Using these forms for the source terms the metric components can be computed, where

(51) |

This solution is interesting primarily because it represents an exact solution which is very difficult to obtain in the presence of higher curvature terms in higher dimensions. Although Eq. ( resembles the de Sitter/anti-de Sitter Reissner-Nordström metric in general relativity, there are several differences. First, the ADM mass and the tidal charge parameter have completely different physical origin, i.e. has contributions from the non-local effects of the bulk Weyl tensor which does not happen in general relativity. In Eq. (, can assume both signatures while in general relativity is always positive. The cosmological constant arises naturally in these models and owes its origin to the higher curvature terms in higher dimensions. Depending on the relative dominance of and , can be positive, negative or zero, such that the resultant metric is asymptotically de Sitter, anti-de Sitter or flat. Recent cosmological observations of distant Type Ia supernovae and the anisotropies in the cosmic microwave background radiation strongly indicate an accelerated expansion of the universe [101, 102, 103, 104, 105] which can be explained by a repulsive cosmological constant with positive . Therefore, it is essential to explore the ramifications of in various astrophysical situations. In what follows we will investigate the influence of the cosmological constant in the continuum spectrum emitted by the accretion disk around quasars, which exhibit strong curvature effects near the horizon. Note however, in our case the origin of the cosmological constant is more physically motivated.

Again for convenience of future computations we redefine the metric components in terms of the gravitational radius, which for metric Eq. ( is given by , (with ) such that the metric components assume the form,

(52) |

where and .

## 4 Spectrum from the accretion disk around black holes in the brane embedded in bulk gravity

In order to probe the observable effects of higher curvature and higher dimensions we consider the near horizon regime of quasars (which host supermassive black holes at the centre) where deviations from general relativity is expected. The electromagnetic emission from the accretion disk around quasars bears the imprints of the background spacetime and hence can be used as a suitable tool to study the nature of strong gravity. In this section we compute the signatures of higher dimensional gravity in the continuum spectrum emitted by the accretion disk around quasars.

The continuum spectrum of black holes depends not only on the nature of the background spacetime but also on the characteristics of the accretion flow. Depending on the equation of state governing the dark radiation and the dark pressure, the background metric is given by Eq. ( and Eq. ( or Eq. (. For the present work we will approximate the accretion flow in terms of the well established “thin-disk model” [106, 107] where the accreting fluid is asumed to be confined to the equatorial plane of the black hole such that the resultant accretion disk is geometrically thin with ( being the height of the disk at a radial distance ). The azimuthal velocity of the accreting fluid dominates the radial velocity and the vertical velocity , such that, . Therefore, such systems do not harbor outflows. The presence of viscosity reduces the angular momentum of the accreting fluid and generates minimal amount of radial velocity which facilitates slow inspiral and fall of matter into the black hole. The gravitational pull of the black hole is assumed to be much stronger compared to the radial pressure gradients and shear stresses such that the accreting gas falls in nearly circular geodesics.

The energy-momentum tensor associated with the accreting fluid is given by,

(53) |

where, is the stress tensor associated with the geodesic flow ( being the proper density and , the 4-velocity of the accreting fluid), constitutes the stress-energy tensor from the specific internal energy () of the system, represents the energy-momentum tensor evaluated in the local inertial frame of the accreting fluid and is the heat flux relative to the local rest frame. Note that both and are orthogonal to the 4-velocity, such that . In the thin-disk approximation, such that the special relativistic correcions to the local hydrodynamic, thermodynamic and radiative properties of the fluid can be safely neglected. Therefore, the entire heat generated due to viscous dissipation is completely radiated away and the accreting fluid retains no heat. As a consequence, only the z-component of the energy flux vector has a non-zero contribution to the stress-energy tensor. For a more elaborate description of the thin accretion disk model one is referred to[106, 107, 108].

The black hole is assumed to accrete at a steady rate and the accreting fluid is assumed to obey conservation of mass, angular momentum and energy. The conservation of mass is given by,

(54) |

where represents the determinant of the metric whose effect on the spectrum we intend to study and is the surface density of the accreting fluid. The conservation of angular momentum and energy assumes the forms,

(55) |

and

(56) |

respectively, where is the angular velocity, is the specific angular momentum and is the specific energy of the accreting fluid. The flux from the disk is given by where,

(57) |

while the height averaged stress tensor in averaged rest frame is denoted by,

(58) |

The conservation laws can be manipulated such that the flux from the accretion disk is given by,

(59) |

where,

(60) |

Eq. ( is derived by assuming that the viscous stress vanishes at the last stable circular orbit such that the accretion disk truncates at . After crossing the marginally stable circular orbit the accreting matter falls radially into the black hole.

By studying geodesic motion of massive test particles in a given static, spherically symmetric spacetime one can derive the angular velocity , the specific energy and the specific angular momentum in terms of the metric components, such that

(61) |

(62) |

and

(63) |

In Eq. (, represents the radius of the marginally stable circular orbit while and are specific energy and specific angular momentum at . The marginally stable circular orbit is obtained from the point of inflection of the effective potential in which the massive test particles move. Therefore it is obtained from the relation, where is given by,

(64) |

Using Eq. ( and Eq. (, Eq. ( can be simplified to give Eq. ( which can be solved to obtain .

The photons thus generated in the system undergo repeated collisions with the accreting gas such that a thermal equilibrium is established between matter and radiation. Such an accretion disk is therefore geometrically thin but optically thick. Consequently, the disk radiates a Planck spectrum at every radial distance with peak temperature given by where (bringing back the and ) and denotes the Stefan Boltzmann constant. By integrating the Planck function over the disk surface one can compute the luminosity from the disk at an observed frequency , such that,

(65) |

where, denotes the gravitational radius, represents the inclination angle of the disk to the line of sight and is the gravitational redshift factor which relates the modification induced in the photon frequency while travelling from the emitting material to the observer [109]. The gravitational redshift factor is given by,

(66) |

Since the spectrum from the accretion disk is an envelope of a series of black body spectra emitted at different peak temperatures, it is often called a multi-color/multi-temperature black body spectrum. Note that the theoretical spectrum depends chiefly on the component of the metric while the component is required only during the integration of the flux to obtain the luminosity (see Eq. () [108].

### 4.1 Effect of bulk gravity on the emission from the accretion disk

In the present work we are interested in investigating the modifications induced in the continuum spectrum of quasars due to the presence of higher curvature gravity in higher dimensions. The background spacetime is therefore given by Eq. ( and Eq. ( for equation of state , while Eq. ( denotes the background metric when the equation of state is given by .

In Fig. 1 we plot the theoretically derived spectrum from the accretion disk when the equation of state is given by for two different masses of supermassive black holes, namely, (Fig. 1(a)) and (Fig. 1(b)). For each of the masses eight spectra are plotted in Fig. 1 by varying the various metric parameters in Eq. ( which are detailed in Table. 1. In each of the spectra the component is similar to the Schwarzschild spacetime (see Eq. () while the component has several corrections to the Schwarzschild metric (see Eq. (). From Table. 1 it is clear that the spectrum labelled by “1” corresponds to the Schwarzschild scenario although the ADM mass owes its origin to higher dimensions and higher curvature terms in the action. This difference in the origin of mass of the black hole cannot be perceived by an external observer. In spectrum “2” the space-time is still Schwarzschild-like although the mass term in the and components of the metric are not the same. From Fig. 1 it is clear that this change hardly affects the theoretical spectrum. In spectra “3” and “4” the mass term in is same as that of the component while and are simultaneously changed as per Table. 1. Fig. 1 shows that change of has an important effect in the spectrum (since spectra “1” and “3” show deviations) while changing barely has any impact (since spectra “3” and “4” are overlapping). For spectra “5” to “8” we fix and since we have understood their effect on the spectrum. Overlap of spectra “5” and “6” imply that has negligible effect on the spectrum. The variation in spectra “1”, “3” and “5” are chiefly due to the disparity in the values of . However, once is lowered below the spectrum becomes insensitive to the changes. This is inferred from the overlap of spectra “7” and “8”.

Fig. 2 depicts the variation of the theoretically derived luminosity with frequency for black hole masses and when the background spacetime is given by Eq. ( which corresponds to the equation of state . The values of the metric parameters corresponding to the nine spectra illustrated in Fig. 2 are given in Table. 2. Spectra “1”, “4” and “7” corresponds to a constant magnitude of , spectra “2”, “5” and “8” are commensurate with while spectra “3”, “6” and “9” are in tandem with . For each set of constant spectra the cosmological constant is variable according to Table. 2. From the virtual overlap of the spectra with constant but variable , it is quite explicit that the tidal charge parameter has a more significant impact on the spectrum than the cosmological constant . Only for the spectrum with appears to be deviated from its counterparts. Note that we cannot choose the magnitude of arbitrarily large as this is in odds with the cosmological observations [102, 105]. On the other hand if is extremely small, it will hardly affect the spectrum. The magnitude of should therefore be chosen in an optimal range.

Moreover, from a theoretical point of view there are restrictions on the maximum positive value of . This stems from the fact that, unlike anti de-Sitter spacetime, a de-Sitter spacetime has a cosmological horizon which is obtained from the largest solution of in Eq. (. Our region of interest should therefore be confined in the region , i.e., the outer radius of the accretion disk should be within . The fact that the inner radius of the disk truncates at automatically ensures that . With an enhancement in , shrinks while increases, such that for (and ) the two horizons coincide and for higher values of , the horizons disappear leading to the formation of a naked singularity [110]. The presence of slightly modifies with a negative marginally lowering the value as opposed to a positive . Also note that we cannot arbitrarily increase , once again to preserve the cosmic censorship conjecture. In the absence of , the presence of an event horizon requires . On increasing the negative value of , the maximum value of gets marginally lowered (e.g. if ) while the presence of a de Sitter enhances the (e.g. if ). However, no real value of can raise upto . Therefore, for all practical purposes we will confine ourselves to and .

A more stringent constraint on is established from the fact that no stable circular orbit exists for in the absence of the charge parameter [110]. Once again the presence of a negative further lowers while a positive raises this value upto a maximum of . Since our accretion disk truncates at we need to keep the maximum value of well below .

The choice of automatically restricts the maximum extent of the accretion disk.
This is because a positive has a repulsive effect as opposed to the attractive force offered by the central black hole. Therefore, the physically relevant region for accretion is the regime where the attractive force due to the black hole dominates. This is given by the static radius where the attractive force due to the black hole and the repulsive force due to nullify. The v