AdS black hole solution with a dark matter halo surrounded by a cloud of strings

Abstract We derive and analyze a Schwarzschild-like Anti-de Sitter (AdS) (BH) obtained as a static, spherically symmetric solution of Einstein’s equations sourced by a cloud of strings (CoS) and a dark matter (DM) halo modeled by a Dehnen-type density profile. We first study the geodesic motion of massless and massive test particles, emphasizing how the CoS parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and the DM halo parameters $$(\rho _s,r_s)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> influence photon spheres, circular orbits, the BH shadow, and the innermost stable circular orbit (ISCO). We then examine scalar perturbations via the effective potential and the associated quasinormal-mode (QNM) spectra, showing how $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and $$(\rho _s,r_s)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> deform oscillation frequencies and damping rates, thereby affecting stability diagnostics. Furthermore, we investigate the thermodynamics in the extended phase space, deriving the Hawking temperature, equation of state, Gibbs free energy, and specific heat capacity, and establishing a consistent first law and Smarr relation with natural work terms for $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and $$(\rho _s,r_s).$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> We find that the interplay between the CoS and the DM halo produces quantitative and sometimes qualitative changes in both dynamical and thermodynamic properties, including shifts in the Hawking–Page transition and heat-capacity divergences, thereby reshaping the phase structure of Schwarzschild–AdS BHs.