Strong gravitational lensing by $$Sgr A^*$$ S g r A ∗ and $$M87^*$$ M 87 ∗ black holes embedded in dark matter halo exhibiting string cloud and quintessential field

Abstract

Abstract We investigate the strong gravitational lensing phenomena caused by a black hole with a dark matter halo in the presence of cloud string and quintessence. This study examines strong gravitational lensing with two significant dark matter models: the universal rotation curve model and the cold dark matter model. To do this, we first numerically estimate the strong lensing coefficients and strong deflection angles for both the universal rotation curve and cold dark matter models. It is observed that the deflection angle, denoted as $$\alpha _D$$ α D , increases with the parameter $$\alpha $$ α while holding the value of $$2M \gamma $$ 2 M γ constant. Additionally, it increases with the parameter $$2M \gamma $$ 2 M γ while keeping the value of $$\alpha $$ α constant. The strong deflection angle $$\alpha _D$$ α D , for the black hole with a dark matter halo, with parameters $$\alpha $$ α and $$2\,M \gamma $$ 2 M γ , greatly enhances the gravitational bending effect and surpasses the corresponding case of the standard Schwarzschild black hole ( $$A=B=0=\alpha =2M\gamma $$ A = B = 0 = α = 2 M γ ). Furthermore, we investigate the astrophysical consequences through strong gravitational lensing observations, using examples of two supermassive black holes, namely $$M87^{*}$$ M 87 ∗ and $$Sgr A^{*}$$ S g r A ∗ , located at the center of nearby galaxies. It is observed that black holes with dark matter halos in the presence of cloud string and quintessence can be quantitatively distinguished and characterized from the standard Schwarzschild black hole ( $$A=B=0=\alpha =2M\gamma $$ A = B = 0 = α = 2 M γ ). The findings in our analysis suggest that observational tests for black holes influenced by dark matter halos, cloud string and quintessence are indeed feasible and viable.