Effect of gravitational lensing around a black hole in a dark matter halo in the presence of plasma

Although General Relativity (GR) remains the cornerstone theory of gravitation and has passed numerous experimental tests in the weak-field regime—from solar system observations to binary pulsar timing—its suitability in extreme gravitational environments, such as those near black holes and in cosmology, continues to be scrutinized [1, 2]. The quest for a unified theory of quantum gravity, the unexplained nature of dark energy and dark matter, and certain cosmological tensions motivate the exploration of alternative and modified theories of gravity [3, 4]. Such theories often predict deviations from GR in the strong-field regime, which may be probed through astrophysical observations of compact objects, particularly black holes [5, 6].

In particular, black holes are not isolated entities but are embedded within complex astrophysical environments, often dominated by dark matter halos [7, 8]. The presence of dark matter can significantly alter the spacetime geometry around black holes, influencing their observational signatures, such as quasinormal modes, shadows, and accretion dynamics [9–11]. Recent advances in gravitational-wave astronomy from LIGO/Virgo [12, 13] and direct imaging from the Event Horizon Telescope [14, 15] have opened new windows for examining these strong-field modifications. Understanding the interplay between dark matter distributions and black hole spacetimes is thus essential for interpreting current and future observational data [16, 17].

In this work, we investigate the effect of gravitational lensing around a Schwarzschild black hole immersed in a Dehnen-type dark matter halo. Gravitational lensing, the phenomenon in which light rays are deflected from their original paths due to the curvature of spacetime, provides one of the most direct observational probes of gravitational fields across a wide range of scales. Its manifestations can be broadly classified into two regimes: strong-field lensing, which occurs in the immediate vicinity of compact objects such as black holes and neutron stars, and weak-field lensing, observed at larger impact parameters within galaxies and galaxy clusters. Each regime offers unique and complementary constraints on the underlying geometry and potential modifications to general relativity (GR) [18−23].

In astrophysical environments, black holes are surrounded by magnetized plasmas whose frequency-dependent refractive index modifies electromagnetic wave trajectories beyond pure geodesic motion. This plasma-lensing interplay is particularly crucial in modified gravity theories, where deviations from the Kerr metric couple with plasma dispersion to produce distinctive, frequency-dependent lensing signatures. Disentangling these effects requires a precise theoretical framework to interpret data from instruments such as the Event Horizon Telescope (EHT) and future observatories including the ngVLA and SKA, especially since plasma effects can mimic or mask metric modifications [24−27]. The influence of plasma on weak gravitational lensing has been extensively analyzed within a variety of gravitational frameworks [28−42].

We investigate gravitational lensing around a black hole embedded in a dark matter halo in a plasma environment, considering various configurations. The paper is organized as follows. In Section II, we introduce the Schwarzschild black hole surrounded by a dark matter halo. Section III is devoted to the dynamics of massive particles, while Section IV discusses photon motion in a plasma environment. In Section V, we analyze weak gravitational lensing in plasma, and Section VI presents the magnification of gravitationally lensed images. The black hole shadow in plasma is examined in Section VII. Section VIII focuses on parameter estimation, and finally, in Section IX, we summarize our main conclusions.

In realistic astrophysical environments, black holes are not isolated objects but are expected to be embedded in dark-matter-filled galactic structures. To model the influence of such environments, it is necessary to consider black hole spacetimes surrounded by a dark matter halo. In this work, we adopt the Dehnen $ (\alpha,\beta,\gamma) $ density profile, which provides a flexible and widely used description of galactic dark matter distributions [43−46]. This profile allows us to model different halo configurations by appropriately choosing the parameters $ (\alpha,\beta,\gamma) $.

where $ \rho_s $ is a characteristic density, $ r_s $ is a scale radius, and the parameters α, β, and γ control the shape of the profile: α determines the sharpness of the transition between the inner and outer regions, γ is the inner slope ($ \rho\propto r^{-\gamma} $ as $ r\to 0 $), and β is the outer slope ($ \rho\propto r^{-\beta} $ as $ r\to\infty $). In this work, we adopt the Dehnen-$ (1,4,\gamma) $ model (i.e., $ \alpha=1 $, $ \beta=4 $). This choice provides a suitable balance between mathematical tractability and physical accuracy in describing the dark matter halo. With these parameters, the density profile simplifies to:

Following the approach developed in [47], the spacetime geometry of a Schwarzschild black hole immersed in a spherically symmetric dark matter halo can be described in Boyer–Lindquist coordinates by a static, spherically symmetric line element.

The presence of the surrounding dark matter distribution modifies the metric function $ f(r) $ relative to the standard Schwarzschild case. The explicit form of this modification depends on the density profile and structural parameters of the halo, such as the central density and the characteristic radial coordinate. As a result, the combined system provides a more realistic description of the gravitational field in galactic environments, enabling the study of the interplay between the black hole and its dark matter surroundings.

For the particular case of the Dehnen profile with $ \gamma = 0 $, the metric function takes the form [47].

where $ \rho_s $ and $ r_s $ denote the characteristic density and scale radius of the dark matter halo, respectively, and M is the black hole mass.

It is important to note that in the absence of dark matter, i.e., in the limit $ \rho_s \to 0 $, the above expression reduces to

This corresponds exactly to the standard Schwarzschild solution. Therefore, the metric given above can be regarded as a natural generalization of the Schwarzschild spacetime that incorporates the gravitational influence of a surrounding dark matter halo. For simplicity, in the following analysis we set the black hole mass to $ M = 1 $.

Figure 1 illustrates the behavior of the event horizon radius coordinate for a Schwarzschild black hole surrounded by a dark matter halo. It is evident that increasing the halo parameters enlarges the horizon radius. In particular, higher values of $ r_s $ and $ \rho_s $ result in a larger black hole horizon, indicating that the dark matter halo modifies the spacetime geometry, causing the horizon to expand.

Figure 1.  (color online) Horizon radius coordinate as a function of the halo parameters $ r_s $ and $ \rho_s $.

To study the motion of test particles around a Schwarzschild black hole surrounded by a dark matter halo, we adopt the standard Lagrangian formulation. The Lagrangian for a particle moving in a curved spacetime is given by

where $ u^{\mu} $ is the four-velocity of the particle and τ is the proper time.

Because the spacetime is spherically symmetric and stationary, two conserved quantities arise from the timelike and rotational Killing vectors. These correspond to the specific energy $ {\cal{E}} $ and the specific angular momentum $ {\cal{L}} $ of the particle. They are obtained from

Without loss of generality, we restrict the motion to the equatorial plane $ \theta=\pi/2 $. Using the normalization condition $ g_{\mu\nu}u^{\mu}u^{\nu}=-1 $ for massive particles, the radial equation of motion can be written as

This equation enables us to define the potential for massive particles as

For the metric function of a Schwarzschild black hole embedded in a dark matter halo, the potential takes the explicit form:

The behavior of the effective potential as a function of the halo parameters $ r_s $ and $ \rho_s $ is illustrated in Fig. 2. These plots illustrate how dark matter modifies the gravitational potential experienced by a massive particle. They also show the locations of the maximum and minimum points, which correspond to unstable and stable circular orbits, respectively. Furthermore, the influence of the halo parameters becomes apparent when compared with the Schwarzschild black hole (SBH) case ($ \rho_s = 0 $). Increasing either $ r_s $ or $ \rho_s $ causes the maximum and minimum points of the effective potential to shift toward each other.

Figure 2.  Radial dependence of the potential $ V(r) $ on the halo parameters $ r_s $ (left panel) and $ \rho_s $ (right panel).

By analyzing the potential, one can determine the allowed regions of motion, as well as the energy and angular momentum of particles moving on circular orbits. For convenience, we introduce the following notation:

where

Figure 3 shows the dependence of the specific energy on the halo parameters. The minima of the energy curves are marked by filled circles, and their corresponding radial coordinates are explicitly given. The specific energy minima shift to larger radii compared to the SBH case.

Figure 3.  (color online) Radial dependence of the square of the specific energy $ {\cal{E}}^2 $ as a function of $ r_s $ (left) and $ \rho_s $ (right).

Similarly, the square of the specific angular momentum of a particle on a circular orbit is given by

Figure 4 presents its behavior as a function of the halo parameters. The minima of the profile are indicated, along with their corresponding radial coordinates. Similar to the trend observed for the specific energy, as $ r_s $ and $ \rho_s $ increase, the positions of the minima shift toward larger radial distances.

Figure 4.  (color online) Radial dependence of the specific angular momentum $ {\cal{L}} $ as a function of $ r_s $ (left) and $ \rho_s $ (right).

The radius coordinate of the innermost stable circular orbit (ISCO) is one of the most fundamental properties of a black hole spacetime, as it determines the inner edge of accretion disks and plays a crucial role in many astrophysical processes. It is derived from the standard conditions

Solving these equations numerically for the metric under consideration yields the dependence of $ r_{{\rm{ISCO}}} $ on the dark matter halo parameters. The results are presented in Fig. 5, which illustrates how the presence of dark matter can significantly shift the ISCO location relative to the standard Schwarzschild case.

Figure 5.  (color online) Dependence of the ISCO radius coordinate $ R_{\text{ISCO}} $ on the halo parameters $ r_s $ (left) and $ \rho_s $ (right).

In a plasma environment, the Hamiltonian governing photon dynamics takes the following form [48]:

where $ x^{\alpha} $ denote the spacetime coordinates and the effective metric $ \tilde{g}^{\alpha\beta} $ is expressed as

Here, n denotes the refractive index of the plasma; $ p_{\alpha} $ and $ u^{\beta} $ correspond to the photon's four-momentum and four-velocity, respectively. The refractive index is given in Ref. [49]

The electron plasma frequency is defined as $ \omega_{p}^{2}(r) = 4\pi e^{2} N(r) / m_{e} $, where e and $ m_{e} $ are the electron charge and mass, and $ N(r) $ is the electron number density. The photon frequency $ \omega(r) $, as registered by a static observer, follows from the gravitational redshift relation:

The constant $ \omega_{0} $ denotes the photon frequency at infinity ($ f(\infty) = 1 $), equivalent to the photon's energy at spatial infinity, $ \omega_{0} = \omega(\infty) = -p_{t} $ [50]. Propagation within the plasma requires the photon frequency to surpass the plasma frequency, i.e., $ \omega_{p} / \omega \lt 1 $; otherwise, propagation ceases. When $ \omega_{0} \approx \omega_{p} $, the resulting deflection angle α substantially exceeds the vacuum case ($ \omega_{p} = 0 $), satisfying $ \alpha \gg 2R/b $, where b is the impact parameter.

By applying the canonical relation $ \dot{x}^{\alpha} = \partial H / \partial p_{\alpha} $ in conjunction with equations (16) and (18), and restricting motion to the equatorial plane ($ \theta = \pi/2, \, p_{\theta} = 0 $), we obtain the four-velocity components of photons:

Combining equations (21) and (22) yields the phase trajectory of light:

Imposing the null condition $ H = 0 $ for photon trajectories allows this equation to be recast as

where the function $ h^{2}(r) $ is defined in [50]

The radius coordinate $ r_{ph} $ of a circular photon orbit, which constitutes the photon sphere, is determined by solving the critical condition [51]

Substituting equation (25) into (26) produces an algebraic equation determining $ r_{p} $ in the presence of a plasma:

Here, the prime denotes differentiation with respect to r. Analytical solutions for $ r_{ph} $ are generally unavailable for arbitrary $ \omega_{p}(r) $ profiles; therefore, specific simplified cases will be examined below.

First, we consider the case of a homogeneous plasma medium, where the plasma frequency $ \omega_{p}^{2}(r) $ is constant throughout the region. Under this condition, Eq. (27) can be solved numerically, and the results are illustrated in Fig. 6. As observed in the figure, the photon sphere radius coordinate $ r_{ph} $ increases as the dark matter halo parameters $ r_s $ and $ \rho_s $ increase. Additionally, the presence of a plasma medium tends to increase the photon sphere radius coordinate compared to the vacuum case.

Figure 6.  (color online) Radial coordinate of the photon sphere as a function of the dark matter parameters $ r_{s} $ (left panel) and $ \rho_{s} $ (right panel) for different values of the homogeneous plasma frequency.

This subsection investigates photon spheres in the context of a non-uniform plasma distribution. We assume the plasma frequency follows a simple power-law profile, as in Refs. [52–53]

where $ z_{0} $ and $ q \gt 0 $ are free parameters. To explore the fundamental characteristics of this model, we focus on two specific cases:

● $ q = 1 $ with constant $ z_{0} $, which precisely replicates the negative-mass diverging lens model [53].

● $ q = 3 $ with constant $ z_{0} $ represents a profile associated with the stellar surface, based on the Goldreich-Julian (GJ) density.

By combining Equations (27) and (28), we determine the photon sphere radius for an inhomogeneous plasma medium using a numerical scheme. The results are presented in Fig. 7. We find that the halo parameters $ r_s $ and $ \rho_s $ have the same qualitative effect: increasing either leads to a larger photon sphere radius $ r_{ph} $. In contrast, the parameter q exhibits the opposite trend; the $ q=3 $ case yields a trend opposite to that of $ q=1 $.

Figure 7.  (color online) The photon sphere radius coordinate as a function of the dark matter parameters $ r_{s} $ (left panel) and $ \rho_{s} $ (right panel) for different values of the inhomogeneous plasma frequency.

This section investigates the effects of gravitational lensing by a Schwarzschild black hole embedded in a dark matter halo and surrounded by a plasma medium. The analysis employs the weak-field approximation, where the spacetime metric is expressed as a perturbation around the Minkowski background [54]:

where $ \eta_{\alpha\beta} $ is the Minkowski metric, $ h_{\alpha\beta} $ is a small perturbation, and the following conditions hold:

The inverse metric, to first order, is given by $ g^{\alpha\beta} = \eta^{\alpha\beta} - h^{\alpha\beta} $, with $ h^{\alpha\beta} = h_{\alpha\beta} $.

Within this framework, the deflection angle $ \hat{\alpha}_b $ for a light ray with impact parameter b can be derived [54]:

Here, $ N(x^i) $ denotes the number density of plasma particles around the black hole, and $ K_e = 4\pi e^2 / m_e $ is a constant, where $ m_e $ is the electron mass.

In Cartesian coordinates, where $ r^2 = b^2 + z^2 $, the metric perturbations for this specific spacetime configuration are:

where $ \cos^2 \chi = z^2 / (b^2 + z^2) $, and $ n_i $ are the components of the unit radial vector.

The total deflection angle in Equation (31) naturally decomposes into three distinct components, each associated with a specific physical contribution:

Here, $ \hat{\alpha}_1 $ arises from the spatial curvature perturbation, $ \hat{\alpha}_2 $ from the modified time-time component of the metric, weighted by the plasma dispersion, and $ \hat{\alpha}_3 $ from the direct refractive effect of the plasma density gradient.

For the first scenario, we examine a homogeneous plasma medium with a constant plasma frequency, $ \omega_{p}^{2} = \text{const} $. This implies $\dfrac{{\rm d}N}{{\rm d}r}=0$, resulting in a spatially uniform refractive index and the vanishing of $ \hat{\alpha}_3 $. Consequently, the final term $ \hat{\alpha}_3 $ in Eq. (33) is omitted. Performing the integration of Eq. (33) under this simplification yields the following expression for the deflection angle:

Using Eq. (34), we compute the results shown in Fig. 8. The plots indicate that the halo parameters $ r_s $ and $ \rho_s $, as well as the uniform plasma parameter $ \omega_p^2/\omega^2 $, have the same qualitative effect: increasing any of these parameters leads to an increase in the deflection angle compared to the SBH case.

Figure 8.  (color online) Deflection angle in a uniform plasma medium as a function of impact parameter b for different parameter values.

The Singular Isothermal Sphere (SIS) is an effective model for describing gravitational lensing. Originally developed to study the properties of gravitational lenses and galaxy clusters, the SIS represents a spherical gaseous configuration whose central density diverges. Its radial density profile is expressed as [54–55]:

where $ \sigma_{\nu}^{2} $ denotes the one-dimensional velocity dispersion. The corresponding particle number density of the plasma is given by the analytic relation [54–55]:

where $ m_{p} $ is the proton mass and κ is a dimensionless constant typically associated with the dark matter component of the Universe. Substituting Eq. (35) into the definition of the plasma frequency yields:

Using the aforementioned characteristics of the SIS model, the deflection angle $ \hat{\alpha}_{\text{SIS}} $ is given by:

where we introduce an auxiliary plasma constant $ \omega_{c}^{2} $, analytically defined as [55]:

To elucidate the impact of the SIS on photon trajectories, we plot the deflection angle $\hat{\alpha}_{\rm SIS}$ as a function of the impact parameter b in Fig. 9. Similar to the behavior observed in Fig. 8, increasing the parameters $ r_s $ and $ \rho_s $ (dark matter halo parameters) and $ \omega_c^2/\omega^2 $ (non-uniform plasma parameter) increases the photon deflection angle in a non-uniform plasma.

Figure 9.  (color online) The deflection angle $ \alpha_{SIS} $ as a function of impact parameter b for different parameter values.

To illustrate the influence of different plasma distributions, Fig. 10 presents a comparative analysis of the deflection angle for different plasma models, adopting the same physical parameters for consistency. The figure shows that the deflection angle in the SIS model is slightly smaller than that in a uniform plasma.

Figure 10.  (color online) Comparison of deflection angles for different plasma models under identical parameters.

We now investigate the image brightness due to the deflection angle around a Schwarzschild black hole surrounded by a dark matter halo in the presence of both uniform and non-uniform plasmas. The combination of light ray angles can be expressed as follows using the lens equation [54, 56]

where $ D_{\rm{s}} $ is the source-observer distance, $ D_{\rm{d}} $ is the lens-observer distance, and $ D_{\rm{ds}} $ is the source-lens distance, while θ and β are the angular positions of the image and source, respectively. One can obtain β from the above equation as [55]:

with

We consider an Einstein ring with radius coordinate $ R_0 = D_{\rm d} \theta_0 $ when the image appears ring-shaped. Then, the expression for $ \theta_0 $ can be written in the following form [57].

The brightness magnification is given by

with $ I_* $ and $ I_{tot} $ representing the unlensed brightness of the source and the total brightness, respectively. The magnification of the source can be expressed as:

Here, $ x=\beta/\theta_0 $ is the dimensionless quantity [55]. The subscript $ \pm $ refers to the location of the image with respect to the source and lens, while the superscript $ \rm{pl} $ represents the presence of the plasma. Finally, using the above equations, we can find the total magnification in the following form

In the subsequent subsections, we explore the magnification for the uniform and non-uniform plasma cases.

We now rewrite Eq. (46) for the uniform plasma as follows [21]

where

with

In Fig. 11, we plot the total magnification as a function of the impact parameter for different values of the spacetime parameters and plasma frequency, in the presence of the uniform plasma. The figure shows that the total magnification increases with the spacetime parameters and the plasma frequency, compared to the SBH case, and also increases slightly with the impact parameter.

Figure 11.  (color online) The plot shows the total magnification as a function of the impact parameter for different values of the spacetime parameters and uniform plasma frequency.

In this subsection, we analyze the total magnification for the non-uniform plasma case. We perform the same calculations as in the previous case. The total magnification for the non-uniform plasma can be written in the following form

where

and

Figure 12 shows the dependence of the total magnification on the impact parameter for different values of the spacetime parameters and the non-uniform plasma frequency. Similar to the uniform case, the total magnification increases as the spacetime parameters and the non-uniform plasma frequency increase. It is worth noting that, in the SBH case, the total magnification is almost independent of the impact parameter, whereas it increases slightly with the impact parameter for the other spacetime parameters. For greater clarity, Fig. 13 presents the magnification ratio for both uniform and non-uniform plasmas relative to the vacuum case. All curves converge to a single point as $ x_0 $ approaches zero. The figure also includes a comparison of the results for the two plasma configurations.

Figure 12.  (color online) The plot illustrates the dependence of the total magnification on the impact parameter for various values of the spacetime parameters and the non-uniform plasma frequency.

Figure 13.  (color online) The plot shows the magnification ratio of uniform and non-uniform plasmas to vacuum for various plasma frequency values.

We now investigate the shadow radius coordinate of a Schwarzschild black hole embedded within a dark matter halo and surrounded by a plasma medium. The angular radius coordinate of the black hole shadow, denoted $ \alpha_{sh} $, can be derived using a geometric approach, leading to the expression [50, 58]:

where $ r_{0} $ and $ r_{ph} $ denote the positions of the observer and the photon sphere, respectively.

If the observer is situated at a sufficiently large distance from the black hole, the shadow radius coordinate can be approximated; see [50]

This approximation follows from the asymptotic behavior $ h(r) \to r $ at spatial infinity, as implied by Eq. (25) for the considered plasma models.

In the vacuum limit, where $ \omega_{\rm p}(r) \equiv 0 $ and $ r_{\rm ph} = 3M $, Eq. (54) reduces to the well-known Schwarzschild shadow radius coordinate:

The shadow radius coordinate in a homogeneous plasma is presented in Fig. 14. We observe that the halo parameters increase the shadow radius coordinate, whereas the plasma reduces it.

Figure 14.  (color online) Shadow radius coordinate in a uniform plasma as a function of the parameters.

The shadow radius coordinate in a nonhomogeneous plasma is shown in Fig. 15. The impact of the halo parameters is similar to that in the homogeneous plasma model. Additionally, the plasma effect becomes negligible when the plasma parameter is $ q = 3 $.

Figure 15.  (color online) Shadow radius coordinate in a non-uniform plasma as a function of the parameters.

We employed a Markov Chain Monte Carlo (MCMC) sampling technique, implemented using the $\mathrm{emcee}$[59] Python package, to estimate the mass and plasma parameters of the Schwarzschild black hole within a dark matter halo. This method samples from the posterior probability distribution of the model parameters, conditioned on observational data.

The likelihood function was constructed using the measured angular shadow diameters of $ \text{M87}^* $ and $ \text{Sgr A}^* $ reported by the Event Horizon Telescope (EHT) collaboration, as listed in Table 1. In our model, the theoretical shadow diameter is linked to the radial coordinate of the photon sphere, which is obtained by numerically solving the corresponding null geodesic equation.

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where $ G = 6.67 \times 10^{-11}\ \text{m}^3\ \text{kg}^{-1}\ \text{s}^{-2} $ is Newton's constant, $ c = 3 \times 10^8\ \text{m}\ \text{s}^{-1} $ is the speed of light, and $ D_{{\rm{obs}}} $ is the distance to the black hole (taken as $ 16.5\ \text{Mpc} $ for $ \text{M87}^* $ and $ 8.275\ \text{kpc} $ for $ \text{Sgr A}^* $).

The log-likelihood function is defined as:

where $ \theta_{{{\rm{sh}}}, i}^{{\rm{obs}}} $ and $ \sigma_i $ are the observed shadow size and its uncertainty, respectively, and $ {\boldsymbol{\Theta}} $ denotes the set of model parameters.

According to Bayes' theorem, the posterior distribution is proportional to the product of the likelihood and the prior:

where $ \pi({\boldsymbol{\Theta}}) $ represents the prior probability distribution. We adopted Gaussian priors for the black hole mass, centered on the literature values, and uniform priors for the other parameters within physically plausible ranges.

For $ \text{M87}^* $:

For $ \text{Sgr A}^* $:

The MCMC analysis, whose sampling results are visualized in Fig. 16, yields estimates for the mass, dark matter halo parameters, and plasma coefficients for both $ \text{M87}^* $ and $ \text{Sgr A}^* $. The best-fit parameter values, presented in Table 2, demonstrate consistency with the EHT observational constraints, supporting the physical viability of our plasma-modified gravitational model.

Figure 16.  (color online) The plot shows the estimated BH mass, DM parameters, and plasma frequency derived from EHT data for $ \text{M87}^* $ (left panel) and $ \text{Sgr A}^* $ (right panel).

In this article, we investigate the observational properties of the Schwarzschild BH surrounded by a dark matter halo, and then constrain the spacetime parameters using the EHT results. We summarize our main results as follows:

● First, we investigate the structure of the event horizon of the Schwarzschild BH surrounded by a dark matter halo. We find that increasing the halo parameters expands the event horizon radius; that is, larger parameter values yield a larger horizon.

● Furthermore, using the Lagrangian formalism, we explore the motion of a massive particle around a Schwarzschild BH surrounded by a dark matter halo. We derive an analytic expression for the effective potential and plot its radial dependence, indicating the maximum and minimum values for different spacetime parameters. We find that the effective potential decreases as the spacetime parameters increase. Subsequently, by analyzing the effective potential, we derive analytic expressions for the specific energy and angular momentum of a massive particle in circular motion and examine their radial profiles. Additionally, by imposing the relevant conditions, we determine the ISCO radii. We find that the ISCO radius increases with increasing dark matter halo parameters.

● Moreover, we study photon motion around the BH in the presence of homogeneous and inhomogeneous plasmas using the Hamiltonian formalism. We determine the photon sphere radial coordinate for each case and plot it as a function of the spacetime parameters. We find that the photon sphere radii increase with the spacetime parameters and the homogeneous plasma frequency. In the inhomogeneous case, the spacetime parameters have the same effect, whereas the plasma slightly increases the photon sphere radii.

● In addition, we study gravitational weak lensing by a Schwarzschild black hole surrounded by a dark matter halo in the presence of uniform and non-uniform plasma. Because of the complexity of the spacetime, we use numerical calculations to analyze the influence of the spacetime parameters and plasma frequency on the deflection angle in each case. We find that the deflection angle increases with the spacetime parameters and plasma frequency in both the uniform and non-uniform plasma scenarios. To provide a more informative comparison, we fix the remaining parameters and examine the deflection angles. Moreover, we explore the magnification of the gravitationally lensed image and plot the total magnification as a function of the impact parameter for different values of the plasma frequency and spacetime parameters. We find that the spacetime parameters and plasma frequency increase the total magnification in both cases. Finally, we compare the magnification ratio for the two plasma distributions with that in vacuum.

● Finally, we study the BH shadow using the same formalism employed to investigate photon sphere radii in homogeneous and inhomogeneous plasma environments. The results indicate that the shadow radii decrease as the plasma frequency increases, whereas they increase with the spacetime parameters. We then constrain the spacetime parameters using observational data released by the EHT collaboration.

As the next step, we can extend our study to rotating black holes. On the other hand, one can study strong gravitational lensing and retrolensing.