Periodic Timelike Motion and Gravitational Wave Signatures around a Magnetically Charged Black Hole Surrounded by Quintessence

Abstract: We investigate timelike geodesics and gravitational wave signatures of periodic motion around a static magnetically charged black hole arising from nonlinear electrodynamics and immersed in a quintessence background. We analyze the effective potential for massive particles, determine the marginally bound and innermost stable circular orbits, and classify the resulting bound trajectories using the zoom-whirl taxonomy $(\mathit{z},\mathit{w},\mathit{v})$. We show that the quintessence parameter $c_q$ systematically shifts the orbital radii, conserved quantities, and turning-point structure associated with representative periodic families. We then model the gravitational radiation emitted by periodic extreme-mass-ratio inspirals within the numerical kludge approximation. The resulting waveforms exhibit the characteristic burst-like structure of zoom-whirl motion, while variations in the quintessence coupling parameter modify the phase evolution, burst timing, and harmonic content of the signal. The corresponding Fourier spectra display a discrete comb-like structure, and the characteristic strain is concentrated in the millihertz band relevant for space-based detectors such as LISA. These results indicate that a quintessence background can leave systematic imprints on periodic orbit dynamics and on the associated time and frequency-domain gravitational wave observables.