Real Part Emergence in Purely Imaginary Quasinormal Modes in Perturbed de Sitter Braneworlds

For braneworlds with infinite extra dimensions, an analysis of the stability of the characteristic spectrum is essential for understanding their dynamical properties. In this study, we investigate the stability of the gravitational perturbation spectrum in a thick de Sitter brane. Unlike the flat brane case, the de Sitter brane features purely imaginary quasinormal frequencies, corresponding to time-domain signals that decay without oscillation. Our results demonstrate that, upon introducing perturbations on the brane, the originally purely imaginary modes develop a nonvanishing real part that depends on the perturbation parameters, thereby becoming complex-frequency modes with both real and imaginary components. In the time domain, this behavior manifests as transient oscillatory signatures in the intermediate stage of the signal, whose fitted frequencies are consistent with those of the first newly induced quasinormal mode, while the late-time waveform remains dominated by the zero mode. As early-time signals are more readily observable, such perturbation-induced oscillations are more likely to be detectable and may have an impact on the extraction of the cosmological constant on the brane from gravitational signals.