Periodic orbits and gravitational waveforms of spinning particles in nonlocal Gravity

In this paper, we investigate the dynamics and gravitational-wave signatures of periodic orbits of spinning test particles moving in the equatorial plane around static, spherically symmetric black holes within the framework of Deser-Woodard nonlocal gravity. Based on the Mathisson-Papapetrou-Dixon equations, combined with the Tulczyjew spin supplementary condition, we derive the orbital dynamic equations for spinning particles moving in the equatorial plane and impose a timelike constraint to exclude unphysical superluminal trajectories. By comparing with the classical Schwarzschild black hole, we systematically analyze the effects of the nonlocal gravitational parameters $\zeta$ and $b$ on the effective potential governing the radial motion of particles and the innermost stable circular orbit. In addition, gravitational waveforms exhibit significant phase differences: an increase in $\zeta$ induces a phase delay, whereas an increase in $b$ results in a phase advance. A one-year simulation of the orbital evolution of an extreme mass ratio inspiral demonstrates that when $b=2$ and $\zeta\approx10^{-6}$, the mismatch between the gravitational waveforms predicted for the nonlocal gravity black hole and those for the Schwarzschild black hole reaches the distinguishable threshold ($\mathcal{M}=0.0125$), providing a basis for observational discrimination between general relativity and nonlocal gravity.