1. Introduction
Even though Einstein's general relativity (GR) is a well-justified and tested theory it is unable to provide potential explanations and insights into the inevitable occurrence of singularities inside black holes (BHs), dark matter (DM; which has not been directly detected) and the accelerated expansion of the Universe. From this perspective, GR has been considered an incomplete theory. However, the existence of BHs in the Universe, predicted by GR, has been proven by recent observations, such as gravitational waves (GWs) [1, 2] and the BH shadow observed through the Event Horizon Telescope (EHT) [3, 4]. These observations play a pivotal role in testing the unique aspects and features of spacetime geometry in the strong-field regime. It is worth noting, however, that promising alternative theories have been proposed to provide a fundamental understanding of the issues faced by GR.
For example, the existence of DM was proposed by initially realizing the flat rotation curves of giant elliptical and spiral galaxies. It is widely believed, based on astrophysical data, that the rotational velocity of stars around giant spiral galaxies can only be explained with the help of elusive DM. It is also worth noting that DM can contribute to as much as around 90% of the mass of the Galaxy, while the rest is the luminous matter made up of baryonic matter [5]. It is also believed that, in the initial evolution of the early Universe, DM was found in regions close to the center of galaxies for the formation of stars. However, in the late stages of galaxy evolution, DM has gradually shifted out to form a DM galactic halo around the host galaxy through different dynamical scenarios. Recent astrophysical observations have reported that almost all giant elliptical and spiral galaxies have a supermassive BH located at the galactic center with a giant DM halo [3, 4]. With these considerations, there have been various approaches to BH solutions involving a DM distribution in the background geometry. For example, Kiselev proposed a static and spherically symmetric BH solution with a DM profile through a quintessential scalar field [6]. Later on, Li and Yang [7] approached this issue in a different way and eventually obtained a similar BH solution with a DM profile represented through a phantom scalar field that contains a term $({r}_{q}/r)\mathrm{ln}(r/{r}_{q}),$ with the boundary r = rq of the DM halo. It is worth noting, however, that the phantom scalar field in this model is not considered in a cosmological role. Instead, it can be modified by involving electric charge and the cosmological constant [8], as well as spin [9]. Following this model, extensive analysis has since been done in a variety of contexts in the literature [10–16]. The spherically symmetric BH solution embedded in a DM halo can also be modeled using a Dehnen-(1, 4, 0)-type DM halo [17]. It is worth noting that this phenomenological Dehnen density profile of DM has been widely examined for galaxies (see, e.g., [18]). In this paper, we consider a Schwarzschild BH embedded in a Dehnen-(1, 4, 0)-type DM halo with a string cloud background and examine its unique aspects through the astrophysical processes occurring around the BH.
In an astrophysical scenario, it is increasingly important to explain the unique aspects and nature of the surrounding fields, such as DM, that can even influence the geodesics of massless particles, causing observable properties such as the BH shadow to be affected. The key point to note is that obtaining a realistic image of a BH is a fundamental question, regardless of the fact that this has been widely investigated in past decades. It is worth noting, however, that there has been much progress in observational studies, such as the recent detection of the first image of a BH at the center of the M87* galaxy (see, e.g., [3, 4]), addressing the existence of BHs in the Universe. BHs have since taken center stage and have been tested by various theories of gravity. In this regard, the BH shadow plays an important role for observers examining the geometry, especially very close to the BH horizon. This results in a dark disc because the light cannot escape from the pull of the BH due to the strong deflection. It must be noted that a dark disk was regarded as a BH shadow by Synge [19] and Luminet [20], addressing light deflection around a spherically symmetric BH. Since then, BH shadows as dark disks have been theoretically modeled and widely examined [21–33]. Recently, a thin accretion disk with relativistic fluid spheres was also tested extensively using observational studies [34]. We note that the unique aspects and nature of the modified gravity theory have also been tested using BH shadows in the context of both rotating and non-rotating BHs [35–37]. The optical properties of charged BHs have also been considered within the Einstein–Maxwell–scalar theory [38]. Additionally, BH shadows were also considered to obtain restrictions on theoretical models using the recent EHT observations [39].
This paper investigates a Schwarzschild BH embedded in a Dehnen-(1, 4, 0)-type DM halo [17] in the background of a cloud string. We shall look into the shadow of this BH since particle motion around BHs encodes information not just about the inherent spacetime geometry but also about the surrounding matter field. This necessitates studying the BH shadow in order to understand the effects of DM and a string cloud. The BH shadow is essentially a black region in the celestial plane surrounded by a ring of light emission. Because BH shadows are dependent on BH parameters they are an interesting area of study. Furthermore, we will calculate quasinormal modes (QNMs), a type of GW produced by the merger of BHs during the ringdown phase [40], with a complex characteristic frequency known as ‘quasinormal frequencies’ (QNFs). The real components of QNFs represent the oscillation frequencies of the perturbation, while the imaginary parts correspond to the decay time. When a BH is disrupted, the relaxation can be represented as a superposition of exponentially damped sinusoidal signals known as QNMs [41]. Thus, during the ringdown stage of the coalescence of two astrophysical BHs, the GWs take the form of superposed QNMs from the remaining BH. According to the no-hair theorem, the frequencies and decay rates of these QNMs are dictated only by the physical properties of the final BH. The measurement of QNMs from GW observations would allow us to test GR and investigate the nature of remains from compact binary mergers.
The paper is structured as follows: section 2 provides a concise description of a Schwarzschild BH embedded in a Dehnen-type DM halo in the background of a cloud string. Section 3 examines the QNMs of a BH disturbed by fields with different spins and derives QNFs. In section 4 , we investigate the BH shadow and study the impact of the core density of the DM halo parameter and the cloud string parameter on the photon sphere and shadow radius. Finally, our conclusions are presented in section 5 .
2. Review of BH metric geometry
In this study, we will investigate the formation of a static BH with a Dehnen-type DM halo in the presence of a cloud string. First, the density profile of the Dehnen DM halo is a specific example of a double power-law profile defined by [42]5 ) the metric function F(r) is related to the tangential velocity through the expression9 ) as a function of r. The plot shows that the center density of the DM halo has a major influence on the occurrence of BH horizons while the core radius and the cloud string parameter remain constant at rs = 0.5 = a. The horizons can be found using the constraint f(r) = 0. Accordingly, there exists a unique horizon for the combination of a DM halo and cloud string.
$\begin{eqnarray}\rho ={\rho }_{s}{\left(\displaystyle \frac{r}{{r}_{s}}\right)}^{-\gamma }{\left[{\left(\displaystyle \frac{r}{{r}_{s}}\right)}^{\alpha }+1\right]}^{\tfrac{\gamma -\beta }{\alpha }}.\end{eqnarray}$
Here $\rho$s is the central halo density, rs is the halo core radius and γ determines the specific variant of the profile. The value of γ lies within [0, 3]. For example, γ = 3/2 is used to fit the surface brightness profiles of elliptical galaxies and closely resembles the de Vaucouleurs r1/4 profile [43]. In this paper, we use the parameters $\left(\alpha ,\beta ,\gamma \right)=\left(1,4,0\right)$ [17]. Therefore the above equation becomes $\begin{eqnarray}{\rho }_{D}=\displaystyle \frac{{\rho }_{s}}{{\left(\tfrac{r}{{r}_{s}}+1\right)}^{4}}.\end{eqnarray}$
We can obtain the mass distribution at any radial distance as $\begin{eqnarray}{M}_{D}=4\pi {\int }_{0}^{r}\rho \left({r}^{{\prime} }\right){r}^{{\prime} 2}{\rm{d}}{r}^{{\prime} }=\displaystyle \frac{4\pi {\rho }_{s}{r}_{s}^{3}{r}^{3}}{3{\left(r+{r}_{s}\right)}^{3}}.\end{eqnarray}$
It is possible to calculate the tangential velocity of a test particle moving in the DM halo from the mass distribution of the halo profile in spherically symmetric spacetime. Thus $\begin{eqnarray}{v}_{D}^{2}=\displaystyle \frac{{M}_{D}}{r}=\displaystyle \frac{4\pi {\rho }_{s}{r}_{s}^{3}{r}^{2}}{3r{\left(r+{r}_{s}\right)}^{3}}.\end{eqnarray}$
We assume the following for a static spherically symmetric line element describing a pure DM halo: $\begin{eqnarray}{\rm{d}}{s}^{2}=-F\left(r\right){\rm{d}}{t}^{2}+\displaystyle \frac{{\rm{d}}{r}^{2}}{F\left(r\right)}+{r}^{2}\left({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}\right).\end{eqnarray}$
In equation ( $\begin{eqnarray}{v}_{D}^{2}=r\displaystyle \frac{{\rm{d}}}{{\rm{d}}r}\left(\mathrm{ln}\sqrt{F(r)}\right).\end{eqnarray}$
Solving for F(r) we obtain $\begin{eqnarray}F(r)=\exp \left[-\displaystyle \frac{4\pi {\rho }_{s}{r}_{s}^{3}\left(2r+{r}_{s}\right)}{3{\left(r+{r}_{s}\right)}^{2}}\right]\approx 1-\displaystyle \frac{4\pi {\rho }_{s}{r}_{s}^{3}\left(2r+{r}_{s}\right)}{3{\left(r+{r}_{s}\right)}^{2}},\end{eqnarray}$
where the leading-order terms of the equation are retained. The next step is to combine the DM profile encoded in F(r) with the BH metric function. We used Xu et al formalism [44], which has also been used by others [17, 45–48], to write the metric line element in this scenario. Thus, the metric of a static and spherically symmetric BH embedded in a DM halo exhibiting a string cloud is described by a line element [17, 49] $\begin{eqnarray}{\rm{d}}{s}^{2}=-f\left(r\right){\rm{d}}{t}^{2}+\displaystyle \frac{{\rm{d}}{r}^{2}}{f\left(r\right)}+{r}^{2}\left({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}\right),\end{eqnarray}$
where $\begin{eqnarray}f\left(r\right)=1-a-\displaystyle \frac{2M}{r}-\displaystyle \frac{4\pi {\rho }_{s}{r}_{s}^{3}\left(2r+{r}_{s}\right)}{3{\left(r+{r}_{s}\right)}^{2}},\end{eqnarray}$
where a refers to the cloud string parameter. Figure 1 shows a visualization of the lapse function (
Figure 1. The function f(r) plotted for different values of the DM central density $\rho$s by setting the core radius; the cloud string parameter is fixed. |
3. Quasinormal modes
In this section we compute the QNMs belonging to a Schwarzschild BH embedded in a Dehnen-type DM halo exhibiting the string cloud spacetime provided in equation (8 ). We use the most standard method, the sixth-order Wentzel–Kramers–Brillouin (WKB) approximation [50–52]. We shall investigate the perturbation equations for three types of perturbation fields with varying spins: a massless scalar field (S), an electromagnetic field (E) and a gravitational field (G). The field equations of the scalar Φ and electromagnetic Aμ field can be described as follows:12 ) are13) –(15 ) to explore both the effect of the DM halo density as well as the cloud string parameter. The figures show that both the core density of the DM halo and the cloud string parameter have a considerable effect on the potentials. Potentials decrease when $\rho$s and a increase. However, the cloud string parameter has more influence than the DM halo.
$\begin{eqnarray}\displaystyle \frac{1}{\sqrt{-g}}{\partial }_{\mu }\left(\sqrt{-g}{g}^{\mu \nu }{\partial }_{\mu }{\rm{\Phi }}\right)=0,\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{1}{\sqrt{-g}}{\partial }_{\mu }\left({F}_{\alpha \beta }{g}^{\alpha \nu }{g}^{\beta \mu }\sqrt{-g}\right)=0,\end{eqnarray}$
where Fαβ = ∂αAβ − ∂βAν is the electromagnetic tensor. The general master equation of a static symmetric BH for calculating QNMs is $\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{2}{\rm{\Psi }}}{{\rm{d}}{r}_{* }^{2}}+\left({\omega }^{2}-V({r}_{* })\right){\rm{\Psi }}=0.\end{eqnarray}$
We will assume the perturbations depend on time as e−iωt. Thus, to have damping, ω must have a negative imaginary part. The tortoise coordinate r* is defined by ${r}_{* }=\int \tfrac{{\rm{d}}r}{f}$. The effective potentials V(r*) for three types of perturbation fields in equation ( $\begin{eqnarray}{V}_{S}(r)=\left(\displaystyle \frac{1}{2}+l\right)\displaystyle \frac{f}{{r}^{2}}+\displaystyle \frac{{{ff}}^{{\prime} }}{r},\end{eqnarray}$
$\begin{eqnarray}{V}_{E}(r)=\left(\displaystyle \frac{1}{2}+l\right)\displaystyle \frac{f}{{r}^{2}},\end{eqnarray}$
$\begin{eqnarray}{V}_{G}(r)=\left(\displaystyle \frac{1}{2}+l\right)\displaystyle \frac{f}{{r}^{2}}-2f\displaystyle \frac{1-f}{{r}^{2}}-\displaystyle \frac{{{ff}}^{{\prime} }}{r},\end{eqnarray}$
where l is the standard spherical harmonic index. Figures 2 and 3 show graphs of the potentials (
Figure 2. Behaviors of BH potential with respect to radial distance r for different values of central density of the DM halo $\rho$s. Here M = 1, l = 2 and rs = 0.5 = a. |
Figure 3. Behaviors of BH potential with respect to radial distance r for different values of the cloud string parameter a. Here M = 1, l = 2 and rs = 0.5 = $\rho$s. |
The next step is to compute the QNFs of the Schwarzschild BH embedded in a Dehnen-type DM halo with a string cloud. To achieve physical consistency at both the BH horizon and infinity, equation (12 ) needs to be subject to the necessary boundary conditions. To have asymptotically flat spacetime, the following quasinormal criteria must be met:
$\begin{eqnarray}{\rm{\Psi }}\left({r}_{* }\right)\to \left\{\begin{array}{cc}{A{\rm{e}}}^{{\rm{i}}\omega {r}_{* }}, & {r}_{* }\to -\infty \\ {B{\rm{e}}}^{-{\rm{i}}\omega {r}_{* }}, & {r}_{* }\to \infty \end{array}\right..\end{eqnarray}$
The coefficients A and B represent the wave amplitudes. These ingoing and outgoing waves are consistent with the physical requirements that nothing can leave the BH’s horizon and no radiation can arise from infinity. Furthermore, these guarantee the existence of an infinite set of discrete complex numbers, often known as QNMs.We focus on the fundamental mode with l = 2 and overtone number n = 0 due to its dominant ingredient of GWs in order to study the impact of Dehnen-type DM parameters and the cloud string parameter on the QNFs. As seen in tables 1 and 2, the frequencies derived by the sixth-order WKB approximation have negative imaginary parts, demonstrating stability under various perturbations.
Table 1. Variation of amplitude and damping of QNMs with respect to the central halo density parameter $\rho$s (for rs = 0.5, a = 0.5, n = 0 and M = 1). |
| $\rho$s | Scalar | EM | Gravitational |
|---|---|---|---|
| 0 | 0.168842 − 0.02412i | 0.164225 − 0.023904i | 0.133632 − 0.022988i |
| 0.2 | 0.15435 − 0.021927i | 0.15016 − 0.021729i | 0.122193 − 0.020899i |
| 0.4 | 0.141975 − 0.02008i | 0.13813 − 0.019902i | 0.112407 − 0.019144i |
| 0.6 | 0.13130 − 0.018513i | 0.12775 − 0.018348i | 0.103965 − 0.017651i |
| 0.8 | 0.12202 − 0.017165i | 0.11874 − 0.017013i | 0.096628 − 0.016368i |
| 1 | 0.11390 − 0.015996i | 0.11084 − 0.015855i | 0.090204 − 0.015254i |
Table 2. Variation of amplitude and damping of QNMs with respect to cloud string parameter (for rs = 0.5, $\rho$s = 0.5, n = 0 and M = 1). |
| a | Scalar | EM | Gravitational |
|---|---|---|---|
| 0 | 0.397707 − 0.077968i | 0.376752 − 0.076623i | 0.307594 − 0.071715i |
| 0.2 | 0.281252 − 0.049663i | 0.269282 − 0.048969i | 0.219422 − 0.046280i |
| 0.4 | 0.180477 − 0.027806i | 0.174658 − 0.027511i | 0.142158 − 0.026297i |
| 0.6 | 0.097018 − 0.012303i | 0.094911 − 0.012215i | 0.077240 − 0.011830i |
| 0.8 | 0.033862 − 0.003063i | 0.033490 − 0.003051i | 0.027282 − 0.003000i |
Figures 4 and 5 show the relationship of QNM with the central density of the DM halo parameter and the cloud string parameter. According to the figures, QNM amplitude and damping generally increase with the parameters $\rho$s and a. However, the effect of a on the QNF is greater than that of $\rho$s. Figures 6 and 7 illustrate the QNFs in the complex frequency plane. The plots show that the general trend of amplitude and damping of QNMs is that both increase with the parameters $\rho$s and a.
Figure 4. Variation of amplitude and damping of QNMs with respect to the central density of the DM halo parameter for three different perturbations. Here M = 1, a = 0.5 and rs = 0.5. |
Figure 5. Variation of amplitude and damping of QNMs with respect to the central density of the DM halo parameter for three different perturbations. Here M = 1, $\rho$ = 0.5 and rs = 0.5. |
Figure 6. Complex frequency plane for the scalar, EM and gravitation perturbations showing the behavior of the QNFs. Here M = 1, a = 0.5 and rs = 0.5. |
Figure 7. Complex frequency plane for the Scalar, EM and gravitation perturbations showing the behavior of the QNFs. Here M = 1, $\rho$ = 0.5 and rs = 0.5. |
4. Shadow
The BH shadow has been extensively explored in the literature because it provides a good opportunity to test various gravity and BH theories under extreme gravity regimes. By using observational data on the BH shadow radius, scientists have been able to constrain model parameters. In this section, we compute and plot the photon sphere rps and shadow radius Rs to analyze their reliance on the DM central density $\rho$s and cloud string parameter a. We also try to limit the parameter space using observational data from the EHT group.
The motion of light can be found by using the Euler–Lagrange equation as follows:21 ) and (22 ) show how the parameters $\rho$s and a affect photon motion in a Schwarzschild BH embedded in a Dehnen-type DM halo exhibiting a string cloud. In either case it is not possible to obtain analytical expressions for the photon radius or the shadow radius. Therefore, we compute numerical values for photon radius and associated shadow radius in order to gain a better grasp of the impact. Table 3 clearly shows that both radii increase with parameters $\rho$s and a.
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}}{{\rm{d}}\tau }\left(\displaystyle \frac{\partial { \mathcal L }}{\partial {\dot{x}}^{\mu }}\right)-\displaystyle \frac{\partial { \mathcal L }}{\partial {x}^{\mu }}=0,\end{eqnarray}$
where the generalized Lagrangian is given by $\begin{eqnarray}{ \mathcal L }(x,\dot{x})=\displaystyle \frac{1}{2}\,{g}_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }.\end{eqnarray}$
We confine ourselves to the equatorial plane (θ = π/2) and define the two constants of motion as $E=f(r)\dot{t}$ and $L={r}^{2}\dot{\phi }$. Thus, the equation for the effective potential is $\begin{eqnarray}{V}_{{\rm{eff}}}=\displaystyle \frac{f}{{r}^{2}}\left(\displaystyle \frac{{L}^{2}}{{E}^{2}}-1\right).\end{eqnarray}$
The radius of an unstable photon orbit can be obtained using the condition${V}_{{\rm{eff}}}=0=V{{\prime} }_{{\rm{eff}}}$, although the photon sphere radius of a BH is calculated from $\begin{eqnarray}{r}_{{ps}}f^{\prime} ({r}_{{ps}})-2f({r}_{{ps}})=0.\end{eqnarray}$
Therefore, the equation for the photon sphere radius of a BH is given by $\begin{eqnarray}\begin{array}{l}\left(9M-3{r}_{{ps}}(1-a)\right){\left({r}_{{ps}}+{r}_{s}\right)}^{3}\\ \quad +\,4\pi {\rho }_{s}{r}_{s}^{3}{r}_{{ps}}\left(3{r}_{{ps}}^{2}+3{r}_{{ps}}{r}_{s}+{r}_{s}^{2}\right)=0.\end{array}\end{eqnarray}$
From the photon radius we can derive the critical impact parameter or the shadow radius Rs as follows: $\begin{eqnarray}{R}_{s}=\displaystyle \frac{L}{E}=\displaystyle \frac{{r}_{{ps}}}{\sqrt{f({r}_{{ps}})}}.\end{eqnarray}$
Equations (Table 3. Numerical values of rps and Rs. Here M = 1 and rs = 0.5. |
| $\rho$s = 0.1 | $\rho$s = 0.5 | $\rho$s = 1 | ||||
|---|---|---|---|---|---|---|
| a | rps | Rs | rps | Rs | rps | Rs |
| 0 | 3.117 52 | 5.414 47 | 3.6093 | 6.319 38 | 4.263 51 | 7.506 42 |
| 0.1 | 3.467 45 | 6.346 81 | 4.027 03 | 7.425 92 | 4.767 81 | 8.835 91 |
| 0.2 | 3.905 05 | 7.579 85 | 4.549 85 | 8.891 04 | 5.399 06 | 10.5972 |
| 0.3 | 4.4679 | 9.269 11 | 5.222 83 | 10.9006 | 6.211 66 | 13.0143 |
| 0.4 | 5.218 63 | 11.6913 | 6.121 06 | 13.7852 | 7.296 32 | 16.4859 |
| 0.5 | 6.27 | 15.3834 | 7.379 75 | 18.1871 | 8.816 28 | 21.7863 |
| 0.6 | 7.847 51 | 21.5204 | 9.269 28 | 25.512 | 11.098 | 30.611 |
| 0.7 | 10.4773 | 33.1672 | 12.4205 | 39.4281 | 14.9034 | 47.385 |
| 0.8 | 15.738 | 60.9976 | 18.7263 | 72.7165 | 22.5177 | 87.5309 |
Figure 8 illustrates how rps and Rs are influenced by DM core density and the cloud string parameter. The figure illustrates how $\rho$s and a have similar effects on the photon sphere and shadow radius. The existence of Dehnen-type DM and a string cloud causes both rps and Rs to increase.
Figure 8. Dependence of rps and Rs with respect to cloud string a for different values of $\rho$s. Here M = 1 and rs = 0.5. |
Now we consider the geometrical quantity on a celestial plane along the coordinates X and Y given by [53]24 ) can be expressed as follows:
$\begin{eqnarray}X=\mathop{\mathrm{lim}}\limits_{{r}_{0}\to \infty }\left(-{r}_{0}\sin {\theta }_{0}{\left.\displaystyle \frac{{\rm{d}}\phi }{{\rm{d}}r}\right|}_{{r}_{0},{\theta }_{0}}\right),\end{eqnarray}$
$\begin{eqnarray}Y=\mathop{\mathrm{lim}}\limits_{{r}_{0}\to \infty }\left({r}_{0}{\left.\displaystyle \frac{{\rm{d}}\theta }{{\rm{d}}r}\right|}_{{r}_{0},{\theta }_{0}}\right),\end{eqnarray}$
where r0 denotes the distance between the BH and the observer. In two-dimensional geometry governed by the shadow radius, equation ( $\begin{eqnarray}{X}^{2}+{Y}^{2}={R}_{s}^{2}=\eta +{\zeta }^{2}.\end{eqnarray}$
We show how the shadow size varies with core density and cloud string parameter in figure 9.
Figure 9. Profile of shadows by a Schwarzschild BH embedded in a Dehnen-type DM halo with a string cloud for different values of $\rho$s (left) and a (right). Here we have set M = 1 and rs = 0.5. |
We will now explore the astrophysical implications of the obtained theoretical results using the recent EHT observations, which provide upper limits for the BH parameters. With this in mind we can assume that supermassive BHs such as Sgr A⋆ and M87⋆ are static and spherically symmetric BHs in the theoretical model considered here. Additionally, we assume that they support our approach and the assumptions we have made here. We then turn to constrain the lower limits of the parameters $\rho$s and a according to the EHT observational data associated with the angular diameter of the BH shadow θ, the distance D and the mass of the supermassive BHs located at the center of the Sgr A⋆ and M87⋆ galaxies. Based on the observational data, the parameters for M87⋆ and Sgr A⋆ are as follows: ${\theta }_{M{87}^{\star }}=42\pm 3\mu {as}$, D = 16.8 ± 0.8 Mpc between Earth and M87⋆ with mass ${M}_{M{87}^{\star }}=(6.5\pm 0.7)\times {10}^{9}{M}_{\odot }$ and ${\theta }_{{{SgrA}}^{\star }}=48.7\pm 7\mu $, D = 8277 ± 9 ± 33 pc and ${M}_{{{SgrA}}^{\star }}=(4.297\pm 0.013)\times {10}^{6}{M}_{\odot }$ (see details in [3, 4]). Taken together, we estimate the diameter of the BH shadow per unit mass, relying on the following equation:
$\begin{eqnarray}{d}_{{sh}}=\displaystyle \frac{D\,\theta }{M}.\end{eqnarray}$
Taking dsh = 2Rsh into further consideration, we determine the diameter of the BH shadow by defining ${d}_{{sh}}^{M{87}^{\star }}=(11\pm 1.5)M$ for M87⋆ and ${d}_{{sh}}^{{{Sgr}}^{\star }}=(9.5\pm 1.4)M$ for Sgr A⋆. Following the observational EHT data, we examine the parameters $\rho$s and a for both Sgr A⋆ and M87⋆ and demonstrate their upper values in figure 10. It is clearly seen from figure 10 that the upper values of $\rho$s and a from observational data of Sgr A⋆ are smaller than those from M87⋆. We show a comparison between the data of M87⋆ and Sgr A⋆. Therefore, we can deduce that the upper value of the DM parameter $\rho$s decreases as the cloud string parameter a increases. However, the effect of both parameters would be much more prominent on the background geometry. We also show the density plot of the parameters $\rho$s and a by applying the data of M87⋆ and Sgr A⋆ in figure 11. The behavior of the parameters $\rho$s and a is also summarized in figure 11. From observational data of M87⋆ and Sgr A⋆, one can estimate the approximate range of the parameters $\rho$s, rs and a. We demonstrate the range of parameters $\rho$s and rs for various possible cases of the cloud string parameter a in figure 12. As shown in figure 12, the DM density $\rho$s is well distributed in the range 0.5 ≤ rs ≤ 1 around the BH. It should be emphasized that the range of rs can be extended up to its possible small values, but the DM density profile $\rho$s decreases when considering the cloud string parameter a, as depicted in figure 12. It is worth noting, however, that the DM density parameter $\rho$s can take values up to <0.5 and <0.12 for M87⋆ and Sgr A⋆, respectively. With this, we can infer from the theoretical results that these observational data for M87⋆ and Sgr A⋆ are potentially important for the best-fit constraints on the BH parameters such as $\rho$s, rs and a and can help constrain the estimations of the DM profile around astrophysical BHs.
Figure 10. Constraint values of the DM density $\rho$s and cloud string parameter a for M87⋆ and Sgr A⋆. Here we have set M = 1 and rs = 0.5. |
Figure 11. The density plot of the DM parameter $\rho$s and cloud string parameter a using the data for M87⋆ and Sgr A⋆. Here we have set M = 1 and rs = 0.5. |
Figure 12. Constraint values of the DM density $\rho$s and the halo core radius rs for M87⋆ and Sgr A⋆ for various combinations of cloud string parameter a. Here, we note that we have set M = 1. |
5. Conclusion
The combination of a BH and DM in the background of a string cloud can be an exciting physical system, potentially crucial in terms of BH properties such as QNMs and shadow cast. In the first part of this study, we derived the functional form of the BH metric function by examining a Schwarzschild BH immersed in a Dehnen-type DM halo profile with a cloud string. We plotted the metric function versus r for various model parameter values and found that the central density of the DM halo has a considerable influence on the presence of a BH horizon while maintaining the core radius and cloud string parameter constant (rs = 0.5 = a). We showed that the combination of a DM halo and string cloud results in a unique horizon.
We then investigated the effective potential of perturbation equations for three types of perturbation field with varying spins: massless scalar, electromagnetic and gravitational fields. Using the sixth-order WKB approximation, we investigated QNMs of the BH disrupted by the three fields and calculated QNFs. The effects of QNM on the core density of the DM halo parameter and the cloud string parameter for three disturbances were investigated. It was found that the core density of the DM halo parameter and the cloud string have opposing effects on the QNM amplitude and damping. Calculations indicated that QNM amplitude and damping increase with $\rho$s and decrease with a, respectively. Tables 1 and 2 show tabulated QNM data that support this pattern. Both $\rho$s and a clearly have an effect on the QNM frequencies. Due to the fact that GWs have a higher frequency and decay rate with $\rho$s and a for the three perturbations, GWs emitting from BHs embedded in a Dehnen-(1,4,0)-type DM halo in the presence of cloud string will propagate further at smaller values of $\rho$s and smaller values of a.
We also examined how the core density of the DM halo parameter and the cloud string parameter affected the photon sphere and shadow radius. We found that both the cloud string and core density parameters have similar effects on the photon sphere and shadow radius. Both the photon sphere and the shadow radius increase as a consequence of an increase in the values of $\rho$s and a. However, the existence of a cloud string has greater impact on shadow size. Since we have shown that the shadow size is dependent on BH parameters, we can utilize the obtained results to establish constraints on the parameters of a BH embedded in a Dehnen-type DM halo in the presence of cloud strings. Using observational data from M87⋆ and Sgr A⋆, we identified a possible range of constraints on the parameters $\rho$s and rs in the presence of the cloud string parameter a. Our analysis indicates that the DM density $\rho$s can be well distributed within the range of 0.5 ≤ rs ≤ 1 around the BH. However, the minimum range of rs can be narrowed down further when considering the cloud string parameter a. It is also noteworthy that the DM density $\rho$s can fall within the ranges of <0.5 and <0.12 with cloud string parameter a < 0.12 and a < 0.032 for M87⋆ and Sgr A⋆, respectively.
Given this study, it is intriguing to explore the impact of the Dehnen profile in the presence of a string cloud on the weak and strong deflection angles of massive particles; we intend to examine this next in a separate study.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grant No. 11675143 and the National Key Research and Development Program of China under Grant No. 2020YFC2201503.