Novel exact black hole solution in Dehnen $\left(1,4,\frac32\right)$ halo thermodynamics, photon circular motion and eikonal quasinormal modes

Abstract: Dehnen $(1,4,\frac32)$ dark matter halo has been proven to be a valuable model for describing the surface brightness distributions of elliptical galaxies, yet its implications for black hole spacetimes remain largely unexplored. In this work, we construct a novel exact black hole solution embedded in this Dehnen halo and investigate its physical consequences. The influence of the halo on black hole thermodynamics is analyzed through the mass function, entropy, Hawking temperature, heat capacity, and Gibbs free energy, allowing us to assess both local and global thermodynamic stability of the black hole-dark matter system. Our results show that the presence of a Dehnen-type halo not only stabilizes the otherwise thermodynamically unstable Schwarzschild black hole but also induces phase transitions. In addition, we study null geodesics to examine photon motion, the shadow radius and the optical appearance of the system. The dark matter halo modifies the effective potential, leading to observable changes in the photon sphere and the apparent size of the shadow. We also explore the instability of circular null geodesics and its relation to quasinormal modes in an eikonal limit. These findings highlight the significant role of realistic dark matter distributions in shaping both the thermodynamic behavior and the observable signatures of black holes, providing further insight into the interplay between dark matter halos and central black holes in galaxies.