Quasinormal modes and dynamical evolution of scalar fields in the Einstein-bumblebee theory with a cosmological constant

In this paper, we investigate the quasi normal modes (QNMs) of static spherically symmetric black holes in the Einstein-bumblebee gravity model, taking into account the effects of the field mass and the cosmological constant. Through separation of the angular components, the scalar field perturbations outside the black hole are reduced to a purely radial main equation. We calculated the quasinormal modes using the matrix method and the WKB approximation, and also studied the dynamical evolution of the purely radial main equation using the finite difference method in the time domain. The eigenfrequencies of the waveforms from the time-domain evolution are fitted to cross-validate the frequency-domain results. The Lorentz-violating parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> , cosmological constant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> , and scalar field mass <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> affect QNMs. Specifically, increasing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> decreases both real and imaginary parts of the monopole modes, but in the dipole and quadrupole modes, the real part remains nearly unchanged while the imaginary part drops rapidly; rising <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> reduces both parts of QNMs; increasing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> raises the real part and lowers the imaginary part. Time-domain analysis confirms these findings, clarifying how Lorentz symmetry breaking impacts QNMs in de Sitter spacetime.