{'fraction_non_alphanumeric': 0.026378896882494004, 'fraction_numerical': 0.008107799474705949, 'mean_word_length': 4.660848954966602, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Thin accretion disk luminosity and its image around rotating black holes in perfect fluid dark matter

Motivated by the fact that the universe is dominated by dark matter and dark energy, we consider rotating black holes surrounded by perfect fluid dark matter and study the accretion process in thin disk around such black holes. Here, we are interested in how the presence of dark matter affects the properties of the electromagnetic radiation emitted from a thin accretion disk. For this purpose, we use the Novikov-Thorn model and obtain the electromagnetic spectrum of an accretion disk around a rotating black hole in perfect fluid dark matter and compare with the general relativistic case. The results indicate that for small values of dark matter parameter we considered here, the size of the innermost stable circular orbits would decrease and thus the electromagnetic spectrum of the accretion disk increases. Therefore, disks in the presence of perfect fluid dark matter are hotter and more luminous than in general relativity. Finally, we construct thin accretion disk images around these black holes using the numerical ray-tracing technique. We show that the inclinaion angle has a remarkable effect on the images, while the effect of dark matter parameter is small.


Introduction
Black holes are fascinating extreme objects predicted by the theory of general relativity (GR). They are known as the strongest sources of gravitational field in the universe and typically expected to have very high spin and magnetic intensity. Due to such features, black holes are an ideal laboratory for studying both matter and gravity in astrophysical experiments. In recent years, some observational evidence have confirmed the existence of black holes. The first was the discovery of gravitational waves resulting from a binary black hole merger by LIGO and Virgo collaboration [1]. Another extraordinary milestone is the first image of M87* black hole shadow [2]- [3] and recently released image of Sgr A* [4] by the Event Horizon Telescope using Very Long Baseline Interferometry. In addition, the observed electromagnetic spectrum of accretion disks can provide confirmation of the existence of black holes [5]- [7]. These recent achievements provide an enlightening response to the understanding of the theory of GR and the nature of accretion disks around supermassive compact objects in the strong gravity regime near the event horizon and may be considered as a way for testing modified theories of gravity.
It is generally believed that compact objects such as black holes gain their mass from the accretion process. Also, to preserve the high accretion rate around such compact objects, the existence of accretion disk is an essential ingredient. An accretion disk is formed by diffusing material which slowly spiral into central compact object and releases gravitational energy as radiation. Images of accretion disks around black holes have been one of the most interesting research areas in observational astronomy since 1970s. The standard model of geometrically thin and optically thick accretion disks was initially proposed by Shakura and Sunyaev in 1973 [8] and later generalised by Page, Novikov and Thorne [9]- [10] to include general relativity which successfully explains the observed spectral features of astrophysical black holes. Also, for a review on theoretical models of black hole accretion disks, see [11]. The signatures appearing in the energy flux and emission spectra of disks provide us not only with information about central massive object, but also provide a test for modified theories of gravity. So, the study of accretion disks around black holes may be considered as a tool to explore possible deviations of alternative theories from GR. The properties of thin disks in various background space-times have been extensively studied [12]- [34].
As we know, according to the standard model of cosmology the matter-energy content of our universe consist of 68% dark energy and 27% dark matter, while the contribution of baryonic matter is only about 5% of the total matter-energy of the universe. In other words, non-baryonic dark matter makes up 85% of the total mass of our universe [35]. Although there has been no direct detection of dark matter, there are some observational evidence confirming its existence including the asymptotic flatness of rotation curves of giant spiral and elliptical galaxies [36], the observations related to the dynamics of galaxy clusters [37] and also the measurements of cosmic microwave background anisotropies [38]. There are different models of dark matter used to explain large and small structure of the universe [39]- [41]. In most of these theories the dark matter is assumed as a halo around black holes [42]- [43]. Moreover, the quintessential dark matter solutions was initially proposed by Kiselev [44]- [45], and entailed construction of spherically symmetric black hole solutions in the presence of perfect fluid dark matter [46]. The perfect fluid model of dark matter has been also studied in [47]. The extension of black hole solutions in th presence of perfect fluid dark matter to the case of rotating black hole solutions is done in [48] and the behavior of rotational velocity with respect to dark matter parameter and black hole spin have also been investigated. The shadow of rotating black holes in perfect fluid dark matter with and without cosmological constant was studied in [49]- [50]. Also, the analysis of the shadow cast by charged rotating black holes in the presence of perfect fluid dark matter has been done in [51]- [52]. The study of circular geodesics in the space-time of charged rotating black holes in perfect fluid dark matter have also been done in [53].
According to the above discussions, it is natural to explore the effects of dark matter on accretion disk luminosity. Such studies, including the luminosity of accretion disks in the space-time of a static black hole in a dark matter envelope and also of black holes surrounded by dark matter with isotropic/anisotropic pressure, have been investigated in [54]- [56]. Now, in the present work we are going to consider rotating black holes surrounded by perfect fluid dark matter [53] and study the influence of dark matter on the electromagnetic properties of thin disks such as the radiant energy flux, temperature distribution, observed luminosity and also radiative efficiency. The results are compared to those of rotating black holes in the absence of dark matter in GR.
The paper is organized as follows. In section 2, we briefly introduce the rotating black holes in perfect fluid dark matter and explain some of their properties. The geodesic motion of test particles in the spacetime of such black holes is studied in section 3. Employing the Novikov-Thorne model, we investigate the electromagnetic properties of thin accretion disks around rotating black holes surrounded by perfect fluid dark matter in section 4. In section 5 we plot the ray-traced redshifted image and intensity and polarization profile of an accretion disk around rotating black holes in perfect fluid dark matter. Finally, we summarized our results in section 6.

Rotating black holes in perfect fluid dark matter
The line element of a rotating black hole surrounded by perfect fluid dark matter is given by [48,53] where M is the black hole mass, a is the spin parameter and α is the perfect fluid dark matter parameter.
In the absence of perfect fluid dark matter, namely in the limit of α → 0, the above metric reduces to the Kerr metric and for the case of α → 0 with a = 0 reduces to the Schwarzschild solution. Meanwhile, for a = 0 it represents the line element of a static and spherically symmetric black hole surrounded by perfect fluid dark matter obtained in [46]. The black hole horizons can be obtained from equation ∆ = 0. The presence of perfect fluid dark matter does not change the number of horizons and only modifies the location of them. There are two horizons corresponding to inner horizon r − , and outer event horizon r + [53]. The behavior of the event horizon as a function of dark matter parameter for different values of spin parameter is plotted in Figure 1. Note that in the present work we consider α to take positive values which gives the positive energy density of dark matter, while in [48] both positive and negative cases have been discussed. As the figure shows, the event horizon radius decreases with the increase of spin parameter so that for a given value of α, the static perfect fluid dark matter black hole with a = 0 has the largest values of the event horizon. Moreover, we see that for a fixed value of a the event horizon radius initially decreases with increasing dark matter parameter and increases around a critical value of α. This increase in the black hole size may be attributed to the fact that after a critical value, the dark matter contributes to the effective mass of the black hole. Therefore, in what follows we consider a more realistic case in which for values of α less than the critical value, α c , the black hole mass overcomes dark matter distribution.

Geodesic equations
A thin accretion disk is a flattened structure formed by gas particles moving on circular orbits around a compact central object. The electromagnetic properties of the disk depend on the geometry of space-time and thus to investigate them, we first study the geodesic motion of test particles in a generic stationary and axisymmetric space-time. The lagrangian is given by where a dot represents differentiation with respect to the affine parameter and the metric of space-time g µν is as follows Since we assume that the metric coefficients are independent of t and φ coordinates, there are two constants of motion corresponding to the energy and angular momentum per unit rest-mass,Ẽ andL, which are given by From the above equations the t and φ components of the four-velocityẋ µ can be obtain aṡ Now, taking the condition 2L = −1 and using equations (7) and (8) we find where the effective potential is given by For circular orbits in the equatorial plane, θ = π 2 , with conditionsṙ =θ =r = 0 the radial component of the geodesic equation leads to the following relation for the angular velocity [57] Ω ± = −g tφ,r ± (g tφ,r ) 2 − g tt,r g φφ,r g φφ,r , where the upper and lower signs denote co-rotating and counter-rotating orbits, respectively. Then, using g µνẋ µẋν = −1 and equations (5) and (6) one can find the specific energyẼ and the specific angular momentumL of test particles moving in circular orbits as Now, using equations (11)-(13) we find the angular velocity, the specific energy and the specific angular momentum for particles moving in circular orbits around a rotating black hole in perfect fluid dark matter as follows with The effective potential is also given by where in the absence of perfect fluid dark matter, α = 0, coincides with that of Kerr black hole. Moreover, the innermost stable circular orbit, r isco , around the central compact object can be determined from the condition V eff,rr = 0, which is equivalent to equation Substituting equations (15)-(16) and the metric components g tt , g tφ and g φφ into above equation leads to which does not have an analytical solution. We have therefore numerically obtained the ISCO radius for different values of a and α parameters. It is clear that in the case of a = α = 0 the above equation reduces to the Schwarzschild black hole with r isco = 6M . In Figure 2, we have shown the effect of both the spin parameter and dark matter parameter on the radius of innermost stable circular orbit. We see that by increasing the spin and dark matter parameters the ISCO radius decreases, however the rate of such a decrease is larger for the spin parameter.

Thin accretion disk properties around rotating black holes in perfect fluid dark matter
We proceed with a brief review of the thin accretion disk model which was presented by Novikov and Thorne in 1973 [9] with additional formula by Page and Thorne in 1974 [10]. The Novikov-Thorne model is a relativistic generalization of that of the Shakura-Sunyaev [8]. We have listed here some of the main assumptions of this model [57]: • The space-time is stationary, axisymmetric, and asymptotically flat.
• The self-gravity of accretion disk is negligible so that the disk's mass does not affect the background geometry.
• The accretion disk is in the equatorial plane of the central massive object.
• The inner edge of the disk is at the ISCO radius and the disk's particles have Keplerian motion between r isco and outer edge r out .
• The accretion disk is geometrically thin, i.e. its vertical size is negligible compared to its horizontal size, h r.
• The disk is optically thick, i.e. the photon mean free path in disk, l, is negligible compared to its depth, l h.
• The disk is in a steady state, namely the mass accretion rate of the disk,Ṁ , does not change with time.
• Magnetic fields are ignored.
In what follows, using the equations of Ω,Ẽ, andL, which were calculated in the previous section, and the desired metric components, we are going to investigate some of the main properties of thin accretion disks in the presence of perfect fluid dark matter.

Radiant energy flux
According to assumptions of Novikov-Thorne model, conservation laws of energy (∇ µ T tµ = 0), angular momentum (∇ µ T φµ = 0), and rest mass (∇ µ (ρu µ ) = 0) of particles in an accretion disk hold for this model [57]. By using these conservation laws, one can obtain the radiant energy flux as [9]- [10] whereṀ is the mass accretion rate. The behavior of energy flux of an accretion disk around a rotating perfect fluid dark matter black hole for different values of parameter α is plotted in the left panel of Figure 3. It can be seen that with increasing α, the energy flux of the accretion disk also increases. Moreover, the peaks of the energy flux diagram tend towards smaller radii with increasing α, which means that most of the energy flux is emitted from the inner part of the disk. To examine the effect of the spin parameter, we have plotted the energy flux when α = 0.1 for different values of the spin parameter a, in the right panel of the figure.
It is clear, with increasing spin parameter, the energy flux also increases, as one would expect.

Radiation temperature
Assuming that the accretion disks in the Novikov-Thorn model are in local thermal equilibrium, one can conclude that the disk emission is blackbody radiation and can be described by the Stefan-Boltzmann law which connects temperature to the energy flux as Where σ SB is the Stefan-Boltzmann constant.
Here we examine the behavior of disk temperature using the available parameters in Figure 4, where the same features can be seen. In the left panel of figure, disk temperature is plotted for different values of α with a = 0.8. It can be seen that the temperature of the disk increases with increasing α with a constant spin parameter. So, the disks rotating around a black hole in perfect fluid dark matter are much hotter than the disks around Kerr black holes. Also, from the right panel we can see that the temperature of the accretion disk for α = 0.1 rises with increasing spin parameter a, so that the disk around a static perfect fluid dark matter black hole has the smallest temperature. The shift of the peak of the diagrams to the left indicates that the inner parts of the disk become hotter with increasing α and a.

Observed luminosity
Assuming that radiation emitted by the disk has a black body spectrum, the observed luminosity of thin disks is given by [58] L(ν) = 8πh cos γ where h is the Planck constant, k B is the Boltzmann constant, and γ is the disk inclination angle which we will set to zero. ν e is the emitted frequency and is given by ν e = ν(1 + z) where z is the redshift factor and can be calculated by Using the above equations for r in = r isco , we have plotted the spectral energy distribution νL(ν) of a thin disk in Figure 5. Similar to the case of radiant energy flux and disk temperature, one may conclude that an accretion disk around a rotating black hole in the presence of perfect fluid dark matter is getting more luminous by increasing α (left panel), while for a fixed value of α the deviation of L(ν) from that of a static perfect fluid dark matter black hole becomes more pronounced with increasing a (right panel). Also, we see that with increasing dark matter parameter, the cut-off frequencies, for which the maximum luminosity is obtained, shift to higher values.

Radiative efficiency
While the material of the disk spirals slowly towards the center, it releases gravitational energy as radiation. The capability of the central object to convert rest mass into outgoing radiation is known as radiative efficiency , which can be calculated by knowing the specific energy in the ISCO radius In Table 1, we have displayed the numerical results of maximum values of the energy flux, temperature distribution and emission spectra and also radiative efficiency of a rotating black holes in perfect fluid dark matter for different values of spin and dark matter parameters. In this section, we investigate the image of a thin accretion disk around a rotating black hole in perfect fluid dark matter as seen by a distant observer. To this end, we construct the image of the disk around black hole using numerical ray-tracing technique [59]. First, we consider a black hole surrounded by an accretion disk and an observer at a distant D from the black hole with inclination angle i, as displayed in Figure 6. The image plane for distant observer is described with Cartesian coordinates system (X, Y, Z) where (X, Y ) are at the image plane, and the Z-axis is perpendicular to the image plane. Another coordinates system is the Cartesian coordinates (x,y,z) which is centered at the black hole. These two coordinates are related by Far away from the compact object, the spatial coordinates (x, y, z) reduce to usual spherical coordinates in the flat space-time. We first write the photon initial conditions in the image plane of distant observer. Let us consider the photon at the image plane at the position (X 0 , Y 0 , 0) with 3-momentum k 0 = −k 0Ẑ perpendicular to the image plane. The initial conditions for the position of the photon are given by t 0 = 0, Also the orthogonality condition uniquely determines the 4-momentum of the photon: k µ = (∂x µ /∂x α )k α , with k α = (k 0 , 0, 0, −k 0 ) being the photon 4-momentum in Cartesian coordinates, which in the spherical coordinates system is Using the above initial conditions, we integrate the geodesic equation with backward ray-tracing method in time, from point (X 0 , Y 0 , 0) at the image plane of distant observer to emission points where τ is the affine parameter. We integrate the geodesic equations using the Runge-Kutta method in MATLAB.
Given that the quasar X-ray emission has been observed to follow a power law, we consider the radiation intensity profile emitted from the disk surface as follows where ν is the photon frequency in the rest frame and µ is the cosine between the photon 4-momentum and the normal vector of the disk measured by the comoving observer, ω(µ) is the angular-dependence of intensity profile which is assumed to be axially symmetrical for simplicity in [59], where ω(µ) is that mentioned in Chandrasekhar's book [61]. Also, Γ and n denote the photon index and the radial steepness of the profile, respectively. In addition, by considering that the central mass is the rotating Kerr black hole and its accretion disk is optically thick according to Novikov-Thorne model, the authors in [59] have investigated the polarization and gravitational Faraday rotation in detail. For further discussion the reader is referred to [62]- [65]. Since photon polarization at the emission source is calculated at the observer location, most authors use the forward ray-tracing method, while Ref. [59] is based on the backward ray-tracing approach. Also, since for a rotating black hole with non-zero angular momentum, the photon polarization plane rotates due to gravitational Faraday rotation [66], radiation from the disk is partially polarized due to magnetic field in the disk atmosphere. Also, because of Thomson scattering of photons from free electrons in the dense atmosphere of the accretion disk, the degree of polarization varies depending on the change in the disk's inclination angle, so that as it increases the degree of polarization also increases.  In addition, it also represents the intensity of radiation on the accretion disk for intensity and polarization profiles. Also, the length and direction of arrows seen in intensity and polarization profiles show the size and direction of polarization respectively.  As is well known, some relativistic factors such as light bending, gravitational redshift and Doppler effect change the emission source frequency which leads to changes of the observed image, as shown in the figures. Such changes are manifest in the hat-like structures (with high inclination angle) and are caused by gravitational lensing effects of the black hole.
The figures also show Doppler blueshift contributions caused by the rotation of the disk in the left half of the plane which can exceed the overall gravitational redshift due to the presence of black hole, so the corresponding photons have a resulting blueshift (left panels). Also, the intensity distribution is concentrated in a small region near the black hole on the left side of the accretion disk (right panels).
Comparing Figures 7 and 8 for the Kerr space-time without dark matter with Figures 9 and 10 for the rotating black hole in the presence of perfect fluid dark matter shows that the central shadow area of the accretion disk gradually decreases by increasing α. Since the space-time under consideration has nonzero angular-momentum, both gravitational Faraday rotation and Thomson scattering affect the photon polarization. Also, with increasing α the size and orientation of polarization in the right panels of the figures decrease so that the Kerr space-time with α = 0 has the most pronounced change of polarization vector in a small region on the disk near the black hole. Comparing the images for the Kerr black hole at two different inclination angles and also comparing the plots for the Kerr black hole (α = 0) with rotating black holes surrounded by perfect fluid dark matter (α = 0) shows that the inclination angle has a remarkable effect on both the ray-traced redshifted image and the intensity and polarization profiles of an accretion disk, while the effect of dark matter parameter α (for α < α c considered here) is small.

Conclusions
Acretion disks around supermassive black holes are the primary source of gravitational information in strong gravity regimes and a representative of the geometry of space-time around them. The study of luminosity of accretion disks in the space-time of a static black hole surrounded by dark matter and of black holes surrounded by dark matter with isotropic/anisotropic pressure has been done in [54]- [56] respectively. Since astrophysical black holes are expected to be rotating due to the accretion effects, in the present work we have studied thin accretion disk properties around rotating black holes surrounded by perfect fluid dark matter.
The rotating black hole solution for perfect fluid dark matter using Newman-Janis method has been obtained in [48]. Considering rotating black holes in perfect fluid dark matter, we studied the properties of thin relativistic accretion disks by employing the Novikov-Thorne model. First, we obtained the physical quantities of interest, namely the effective potential V eff , specific angular momentumL, specific energỹ E, angular velocity Ω, and ISCO radius for test particles moving in circular orbits around the black hole. Since the ISCO equation does not have an analytical solution, we numerically calculated the ISCO radius for both spin, a, and dark matter, α, parameters. The analysis of the ISCO radius shows that for a given value of α the ISCO radius decreases with increasing spin parameter, while at a fixed spin parameter the ISCO radius decreases up to a critical value of α and then increases with increasing α. For α = 0 it coincides with ISCO radius of Kerr black hole in the absence of dark matter.
To calculate the heat emitted by the accretion disk, we generalized the relations of Ref. [10] to Kerr perfect fluid dark matter black holes and numerically obtained the energy flux F (r), temperature T (r), and luminosity distribution L(ν) and plotted their profiles. The decrease in ISCO radius shows that in the presence of dark matter the strength of gravitational field decreases with α so that the inner edge of the disk, which is located at the ISCO radius in Novikov-Thorn model, becomes close to the event horizon. Therefore, in comparison with rotating black holes in the absence of perfect fluid dark matter, the energy flux, temperature and luminosity over the disk surface increase which is consistent with the results of Table 1. In summary it can be said that thin accretion disks around rotating black holes surrounded by perfect fluid dark matter are hotter and more luminous than Kerr black holes without dark matter. It is therefore plausible to conclude that all the real mass measured by luminosity is not related to baryonic mass. It is, however, possible that some fraction of this luminosity is related to accretion disks around compact objects.
We also investigated accretion disk images around rotating black holes in perfect fluid dark matter using a ray-tracing code written in MATLAB [59]. We plotted the ray-traced redshifted images, intensity and polarization profiles of accretion disks for Kerr black holes in the absence of dark matter and also for rotating black holes surrounded by perfect fluid dark matter. Comparing the corresponding results show that with increasing α the central shadow area decreases while the polarization vector is susceptible to less rotation. Also, the results show that the effect of α is negligible in comparison with the inclination angle.