Black-hole Shadows and Null Geodesics in Hamaus–Sutter–Wandelt Void Spacetimes with a Quintessential Field: Observational Signatures from EHT Data of M87 ^∗ and Sgr A ^∗

Abstract We study horizon-scale lensing and shadow formation for a static, spherically symmetric black hole embedded in a cosmic void modeled by the Hamaus–Sutter–Wandelt (HSW) density profile and immersed in a quintessential field. Starting from the enclosed mass implied by the HSW profile, we derive a metric function f ( r ) that augments the Schwarzschild term by void-dependent contributions and a quintessence term. We obtain analytic small and large r expansions and a global interpolant for f ( r ), then analyze null geodesics to determine the photon sphere and the shadow radius <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo>/</mml:mo> <mml:msqrt> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msqrt> </mml:math> . The photon-sphere condition does not admit a closed form and is solved numerically. We confront the predicted shadows with Event Horizon Telescope (EHT) measurements of M87* and Sgr A*, translating angular diameters, distances, and masses into dimensionless shadow sizes. The comparison yields quantitative bounds on the void and quintessence parameters: δ c ∈ [−0.55, 0] and ρ s ∈ [0.003, 0.012], with γ ∈ [0.005, 0.12] (Sgr A*, 1 σ ), and correspondingly broader intervals for M87* that accommodate slightly larger values. Across the explored space, increasing δ c (toward zero), or increasing γ and ρ s , reduces R s , reflecting a softer effective curvature near the photon sphere. A dynamical-systems analysis of the radial motion shows that circular null orbits remain unstable saddles, with their location shifting monotonically with δ c , γ , and ρ s . Finally, using the limiting absorption cross section <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">lim</mml:mi> </mml:mrow> </mml:msub> <mml:mo>≃</mml:mo> <mml:mi>π</mml:mi> <mml:msubsup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> , we show that the Hawking energy-emission rate is enhanced by decreasing γ and ρ s and suppressed as these increase, while larger underdensities (more negative δ c ) favor higher emission. Our results establish EHT-level, data-driven constraints on HSW quintessence environments and quantify their imprint on black-hole shadows and thermodynamics.

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