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

We found that, unlike General Relativity, the shift-symmetric subclass of Beyond Horndeski theories provides black holes with primary hair, which are thermodynamically stable and compatible with current Event Horizon Telescope observations for M87* and Sgr A* black holes. We first investigate its thermodynamic properties by studying the influence of the primary hair on the thermodynamic quantities and the local stability, imposing severe constraints in the allowed range of the primary hair values. Then, we analyze the null geodesics near this black hole, demonstrating how the scalar hair influences the shadow diameter, finding that higher values of the primary hair lead to smaller shadow sizes. Additionally, we constrain the scalar hair using the observational data, revealing its sensitivity to the choice of other black hole parameters. To further probe the observational features of the scalar hair, we simulate face-on two-dimensional images of spherically infalling accretion disks, examining how the primary scalar hair affects the black hole's shadow. Finally, we impose all relevant constraints to identify black holes that are both stable and consistent with observational data.