# Weak Gravitational Lensing around Bardeen Black Hole with a String Cloud in the Presence of Plasma

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## Abstract

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## 1. Introduction

## 2. Bardeen Spacetime with a Cloud of Strings

## 3. Weak Gravitational Lensing with Plasma

#### 3.1. Uniform Plasma with ${\omega}_{e}^{2}=const$

#### 3.2. Non-Uniform Plasma with Singular Isothermal Sphere Medium

#### 3.3. $Non$-$Singular$ $Isothermal$ $Gas$ $Sphere$

## 4. Magnification of Image Source in the Presence of Plasma

#### 4.1. Uniform Plasma with ${\omega}_{e}^{2}=\phantom{\rule{3.33333pt}{0ex}}\mathrm{const}$

#### 4.2. Non-Uniform Plasma with SIS Medium

## 5. Conclusions

- We analyzed the horizon radius for various values of the black hole in Bardeen gravity with a cloud string parameter a. The results show that the radius of horizon ${r}_{h}$ is decreasing in the presence of parameter q, and the effect of string clouds is inverse.
- In the presence of the parameter a and magnetic charge q of the Bardeen spacetime metric with a cloud of a string field, the deflection angle of light beams around the BH decreases. In addition, with fixed values of the magnetic charge q and parameter a, the influence of uniform plasma on the gravitational weak deflection angle is also shown in Figure 3.
- As seen in Figure 4, the deflection angle of light rays around the compact object increases as the value of the parameter responsible for the non-uniform plasma medium increases.
- The deflection angle of a light beam around the BH is larger in the uniform case than in the non-uniform case, and this is true regardless of the values of the parameters a and q in the plasma (Figure 6).
- For various values of the parameter a and magnetic charge q in Bardeen spacetime in the presence of a cloud string field, we have explored the overall magnification of the image source caused by gravitational weak lensing. We have demonstrated that the magnetic charge $q/M$ has an impact on the total magnification of the image but that the total magnification also increases with an increase in the parameter a (see Figure 8 and Figure 9).
- Finally, we have investigated the total magnification’s reliance on the plasma medium and found that it increases as the plasma medium’s uniform and non-uniform properties are increased.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The separatrix line indicates the border corresponding to extremal black holes, which separates black holes from no black holes.

**Figure 3.**The dependence of the deflection angle ${\widehat{\alpha}}_{uni}$ on the impact parameter b and parameter a, and magnetic charge q for different values of parameter $a/M$, plasma medium ${\omega}_{e}^{2}/{\omega}^{2}$, and magnetic charge q. The corresponding fixed parameter used is b = 4.

**Figure 4.**The effect of the impact parameter b, parameter a, and magnetic charge q on the deflection angle ${\widehat{\alpha}}_{SIS}$ for different values of parameter $a/M$, plasma medium ${\omega}_{c}^{2}/{\omega}^{2}$, and magnetic charge q. The associated fixed parameter is b = 4.

**Figure 5.**Deflection angle ${\widehat{\alpha}}_{\mathrm{NSIS}}$ as a function of impact parameter b, plasma frequency $\frac{{\omega}_{c}^{2}}{{\omega}^{2}}$, and parameter a and magnetic charge q. The fixed parameters used are $b=4$ and ${r}_{c}=3$.

**Figure 6.**Plot of the deflection angle ${\widehat{\alpha}}_{b}$ as a function of the impact parameter b, magnetic charge q, and parameter a. The corresponding fixed parameters used are $b=4$ and ${r}_{c}=3$.

**Figure 7.**Schematic view of the gravitational lensing system (adopted from Ref. [81]).

**Figure 8.**The total magnification of the image brightness in the presence of uniform plasma as a function of ${\omega}_{c}^{2}/{\omega}^{2}$, q, b, and a. The fixed parameters used are ${R}_{s}=2$, $b=3$, and ${x}_{0}=0.055$.

**Figure 9.**The total magnification of the image brightness in the presence of SIS as a function of ${\omega}_{c}^{2}/{\omega}^{2}$, q, b, and a. The fixed parameters used are ${R}_{s}=2$, $b=3$, and ${x}_{0}=0.055$.

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**MDPI and ACS Style**

Atamurotov, F.; Alibekov, H.; Abdujabbarov, A.; Mustafa, G.; Aripov, M.M.
Weak Gravitational Lensing around Bardeen Black Hole with a String Cloud in the Presence of Plasma. *Symmetry* **2023**, *15*, 848.
https://doi.org/10.3390/sym15040848

**AMA Style**

Atamurotov F, Alibekov H, Abdujabbarov A, Mustafa G, Aripov MM.
Weak Gravitational Lensing around Bardeen Black Hole with a String Cloud in the Presence of Plasma. *Symmetry*. 2023; 15(4):848.
https://doi.org/10.3390/sym15040848

**Chicago/Turabian Style**

Atamurotov, Farruh, Husan Alibekov, Ahmadjon Abdujabbarov, Ghulam Mustafa, and Mersaid M. Aripov.
2023. "Weak Gravitational Lensing around Bardeen Black Hole with a String Cloud in the Presence of Plasma" *Symmetry* 15, no. 4: 848.
https://doi.org/10.3390/sym15040848