Thermodynamics, thermodynamic geometry, and the shadow of a Bardeen black hole surrounded by a Kiselev fluid: the role of regularity, asymptotics, and observational self-consistency

Abstract

We reconsider the Bardeen black hole surrounded by a Kiselev fluid, obtained by adding a Kiselev term to the Ayon-Beato-Garcia-Bardeen metric. Contrary to claims in the literature, the solution is regular at the origin only for $c=0$: for $c\ne 0$ curvature invariants diverge as $r\to 0$, and the spacetime is asymptotically de Sitter-like rather than flat, with a cosmological horizon $r_c$ for $0<c<c_N(q)$. We map the full $(q,c)$ domain of horizon structures in closed form, locating the extremal and Nariai-type boundaries and their cusp. Using the Ma-Zhao prescription for the mass-dependent nonlinear-electrodynamics source, we derive the corrected entropy and verify the extended first law and Smarr relation to machine precision by computer-algebra differentiation, not through the entropy's own definition. The heat capacity shows a single Davies-type divergence, located as a closed-form quartic and expanded analytically in $c$. In thermodynamic geometry, the Quevedo-metric curvature has a spurious second singularity generated by the entropy's integration constant and the non-uniqueness of its conformal factor; we isolate this artefact and show only the Davies-point divergence is physical. The Ruppeiner curvature, identically zero for pure Bardeen, becomes non-zero once $c\ne 0$. For the shadow, we compute the critical impact parameter $b_c(q,c,w)$ and show the nominal bound $c\lesssim 10^{-2}/M$ from the EHT image of Sagittarius A* is not self-consistent: the same $c$ places the cosmological horizon nine orders of magnitude closer than Earth's distance to Sgr A*, and the bound is fourteen orders of magnitude larger than $H_0/c$, the true cosmological dark-energy scale. We reinterpret the EHT constraint as bounding $c$ only near the horizon, not as an astrophysical or cosmological bound on the quintessence parameter.

arXiv comment16 pages, 8 figures