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

TL;DR

The paper constructs an exact static black hole solution embedded in a Dehnen $(1,4,\tfrac{3}{2})$ dark-matter halo and analyzes how the halo modifies horizon structure, curvature, and energy conditions. It develops the halo-corrected thermodynamics, revealing locally stable branches and second-order phase transitions driven by halo parameters, and links photon-circle dynamics to both shadows and eikonal quasinormal modes through a Lyapunov-exponent connection. The study shows halo-induced changes to the photon sphere and shadow radius, and provides weak-lensing and deflection-angle results that could inform observational tests with EHT data and future lensing measurements. The eikonal QNM analysis yields frequencies tied to the photon-sphere geometry, with Schwarzschild results recovered in the halo-free limit, offering a consistent bridge between black-hole spectroscopy and realistic galactic environments. Overall, the work demonstrates that realistic dark-matter distributions can leave observable imprints on black-hole thermodynamics and optical signatures, motivating further exploration of DM halos in strong-field gravity.

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.

Figures (15)

• Figure 1: Metric function $g_{tt}$ as a function of $r$ for $r_s=2$. Left: fixed $r_0=0.1$, Right: fixed $\rho_0=0.01.$
• Figure 2: Left: $M_{DM}(r_H)$ as a function of $r_H$. Right: 3D plot of $M_{DM}(r_H)$ as a function of $\rho_0$ and $r_0$.
• Figure 3: Graphical demonstration of three curvature invariants Left: $R$, Middle: $R_{ab}R^{ab}$ and Right: $R_{abcd}R^{abcd}$.
• Figure 4: Four energy conditions as radial function for various value of $\rho_0$ and $r_0$. In the WEC panel, only $\rho_E\geq0$ is shown.
• Figure 5: The mass function plotted against $r_H$ for different core density $\rho_0$ and fixed $r_0=0.5$.