Periodic orbits and gravitational waveforms in quantum-corrected black hole spacetimes

In this paper, we study the periodic orbits of massive particles around two quantum-corrected black holes proposed in effective quantum gravity, and explore the quantum gravity effect on both the particle orbits and the associated gravitational wave signals. First, we analyze the geodesic motion of the massive particle around the black holes. We then study two important types of bound orbits of the massive particles, the marginally bound orbit and the innermost stable circular orbit. We find that, for the first black hole, increasing the quantum parameter $\zeta$ leads to larger orbital radii and reduced angular momenta for both orbits. In contrast, the second black hole shows $\zeta$-independent orbital radii and angular momenta. By analyzing the effective potential, we determine the allowed range of the energy and the angular momentum for bound orbits, with $\zeta$-dependence only for the first black hole. We further investigate periodic orbits with a fixed energy for both black holes, revealing that the parameter $\zeta$ similarly affects the orbits, although its effect is negligible in the second black hole. Finally, we calculate the gravitational waves emitted by the periodic orbits. The results demonstrate that increasing $\zeta$ leads to a significant phase delay for the first black hole, while only inducing a subtle phase advance for the second one. Therefore, we conclude that the first black hole can be distinguished from the Schwarzschild one through gravitational wave observations, whereas the second one cannot be effectively distinguished when the quantum correction is weak.