Observable signature of Dehnen-type dark matter halos via strong gravitational lensing by supermassive black holes and constraints from EHT observations

We investigate strong gravitational lensing by black holes influenced by a Dehnen-type dark matter halo, characterized by an additional parameter, the core radius $r_s$, and the dark matter central density ${\rho }_s$, alongside the black hole mass $M$. Our findings reveal that, for a fixed value of the dark matter central density ${\rho }_s$, key parameters such as the unstable photon orbit radius $r_m$, critical impact parameter $u_m$, angular position ${\theta }_{\infty }$, and angular separation $S$ increase with the growing value of the core radius $r_s$, while the relative magnitude $r_{mag}$ decreases. Conversely, for a fixed value of $r_s$, parameters including the unstable photon orbit radius $r_m$, critical impact parameter $u_m$, strong deflection angle ${\alpha }_D$, angular position ${\theta }_{\infty }$, angular separation $S$, relative magnitude $r_{mag}$, and the outermost Einstein ring ${\theta }_1^E$ increase with the growing value of the dark matter central density ${\rho }_s$. For Sgr A*, the outermost Einstein ring ${\theta }_1^E$ is larger than in the case of M87*. We have calculated the time delay between two relativistic images by utilizing various supermassive black holes in nearby galaxies. It is found that the time delay under the influence of a Dehnen-type dark matter halo ($\sim 33.7584$ min) is suppressed compared to the case of a Schwarzschild black hole ($\sim 38.7682$ min), as in the case of NGC 2778. Finally, we constrain the central density parameter of the dark matter halo, ${\rho }_s$, for a fixed core radius $r_s$, using observational data from the Event Horizon Telescope collaboration for the supermassive black holes $M87^{\ast }$ and $Sgr{A}^{\ast }$. The dark matter halo parameter ${\rho }_s$ is constrained to the range $2.06\le {\rho }_s{M}^2\le 2.67$ for $M87^{\ast }$ and $1.9\le {\rho }_s{M}^2\le 2.26$ for $Sgr{A}^{\ast }$. Our results suggest that black holes enveloped by a Dehnen-type dark matter halo align with EHT observations, making it a promising candidate for explaining the surrounding environment of black holes.