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 R s = r p / f ( r p ) . 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 σ lim ≃ π R s 2 , 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.