Quasinormal Modes and Dynamical Evolution of Scalar Fields in the Einstein-Bumblebee Theory with a Cosmological Constant

This paper investigates the dynamic behavior of static, spherically symmetric black holes within the Einstein-Bumblebee gravity model with a cosmological constant, focusing on scalar field perturbations. Through separation of the angular components, the scalar field perturbations outside the black hole are reduced to a purely radial main equation. The quasinormal modes (QNMs) of the system are then determined via the WKB approximation in the frequency domain, while the dynamic evolution of the system is examined in the time domain using finite difference methods. The eigenfrequencies of the waveforms from the time-domain evolution are fitted to cross-validate the frequency-domain results. The study finds that the Lorentz violation parameter $\ell$ and the cosmological constant $\Lambda$ significantly influence the QNMs. Specifically, as $\ell$ increases, the real and imaginary components of the lower modes decrease, while in higher modes, the real part changes minimally, and the imaginary part decreases rapidly. An increase in $\Lambda$ similarly results in a decrease in the overall QNM values. These results are supported by the time-domain analysis, providing a clearer picture of how Lorentz symmetry breaking affects the QNMs of de Sitter spacetime.