Thermodynamic and observational constraints on black holes with primary hair in Beyond Horndeski gravity: Stability and shadows

In this paper, we find that unlike in General Relativity, the shift-symmetric subclass of Beyond Horndeski theories permits black holes with primary hair that are thermodynamically stable and align with current Event Horizon Telescope observations of the M87* and Sgr A* black holes. This work begins by investigating thermodynamic properties, analyzing how primary hair influences thermodynamic quantities and local stability, which imposes strict constraints on the allowed range of primary hair values. The null geodesics near this black hole are then examined, demonstrating how scalar hair affects the shadow diameter. Specifically, when the parameter of the Beyond Horndeski function F ₄ is negative, increasing scalar hair enlarges the shadow; in contrast, when this parameter is positive, greater scalar hair reduces the shadow size. Further constraints on the scalar hair are derived using observational data, highlighting its sensitivity to other black hole parameters. To explore additional observational features, face-on two-dimensional images of spherically infalling accretion disks are simulated, revealing how primary scalar hair shapes the black hole's shadow. Finally, all relevant constraints are combined to identify black holes that are both stable and consistent with observational data.