Distinguishing Between Dark Matter-Black Hole Systems and Naked Singularities via Quasi-Periodic Oscillations

Quasi-Periodic Oscillations (QPOs) are an important phenomenon commonly observed in the X-ray radiation of black holes and neutron stars, closely related to the dynamics of accretion disks around compact objects and general relativistic effects. The objective of this study is to use the QPO phenomenon to distinguish between dark matter-black hole systems and naked singularities, as well as to investigate the effects of different dark matter models (Cold Dark Matter, CDM, and Scalar Field Dark Matter, SFDM) on the accretion disk dynamics. By introducing a dark matter correction model within the framework of general relativity, we systematically investigate the differences in dragging effects, characteristic frequency distribution, and the innermost stable circular orbit (ISCO) radius between dark matter-black hole systems and naked singularities, while analyzing the potential coupling between QPO frequencies and dark matter distribution. The main results of this study are as follows: $\nu_r$ and $\nu_\theta$ in dark matter-black hole systems can be identified as HFQPOs, while for lower spins ($a < 0.5$), $\nu_\text{nod}$ can be identified as LFQPOs, and for higher spins ($1 > a \geq 0.5$), $\nu_\text{nod}$ falls within the HFQPO observation range. Cold Dark Matter (CDM) and Scalar Field Dark Matter (SFDM) modulate the accretion disk dynamics at the order of $10^{-6}$.