Periodic orbits and gravitational waveforms around a Schwarzschild black hole with a cloud of strings embedded in perfect fluid dark matter

In this study, we explore the dynamics of particle orbits and their corresponding gravitational wave signatures in the vicinity of a Schwarzschild black hole (BH) surrounded by a cloud of strings and embedded in a perfect fluid dark matter medium. The model is characterized by two parameters: $a$, associated with the string cloud, and $\alpha$, representing the dark matter distribution. We systematically analyze how the marginally bound orbit (MBO) and the innermost stable circular orbit (ISCO) depend on these parameters. Our findings reveal that while both the orbital radius and angular momentum increase with increasing $a$, they decrease as $\alpha$ increases; notably, the energy exhibits the opposite trend, decreasing with $a$ and increasing with $\alpha$. Furthermore, we examine periodic orbits indexed by rational numbers $q$ and the gravitational waveforms they generate. The results demonstrate that an increase in the string cloud parameter $a$ induces a significant phase delay in the waveform. Specifically, waveforms with lower values of $a$ oscillate over shorter time intervals, whereas those with higher values extend to longer time scales. These distinct features, including noticeable differences in amplitude, allow the waveforms to be clearly distinguished from those in a pure Schwarzschild spacetime.