Testing black holes with cosmological constant in Einstein-bumblebee gravity through the black hole shadow using EHT data and deflection angle

Abstract: This study probes spacetime solutions within Einstein-Bumblebee gravity, a modified gravitational framework incorporating spontaneous Lorentz symmetry violation through a vector field mechanism. By introducing a cosmological constant into this model, the research scrutinizes thermodynamic properties of black holes in both anti-de Sitter (AdS) and de Sitter (dS) geometries. The investigation demonstrates how Lorentz-violating parameters alter foundational thermodynamic principles, including revisions to the first law of black hole mechanics and shifts in critical phenomena during phase transitions. Notably, the bumblebee coupling parameter emerges as a critical factor governing horizon structure and thermal emission characteristics, with pronounced deviations from general relativity (GR) predictions observed as this parameter increases. The analysis extends to observational signatures by calculating shadow profiles of these modified black holes. Shadow morphology exhibits dual dependence on the cosmological constant and the bumblebee parameter, presenting measurable discrepancies from classical relativity that could be constrained through Event Horizon Telescope (EHT) observational data. Furthermore, using geometric formalisms, the study quantifies light deflection phenomena in weak and strong gravitational regimes. Results reveal that both the cosmological constant and Lorentz-violating parameter induce detectable modifications to lensing angles compared to Schwarzschild or Kerr benchmarks. These deviations, while subtle, underscore the necessity for next-generation astronomical instruments capable of resolving fine-scale spacetime curvature effects.