Distinguishing a Kerr-like black hole and a naked singularity in perfect fluid dark matter via precession frequencies

We study a Kerr-like black hole and naked singularity in perfect fluid dark matter (PFDM). The critical value of spin parameter $a_c$ is presented to differentiate the black hole from naked singularity. It is seen that for any fixed value of dark matter parameter $\alpha$ the rotating object is black hole if $a\leq a_c$ and naked singularity if $a>a_c$. Also for $-2\leq\alpha<2/3$ the size of the black hole horizons decrease whereas for $2/3<\alpha$ it increases. We also study spin precession frequency of a test gyroscope attached to stationary observer to differentiate a black hole from naked singularity in perfect fluid dark matter. For the black hole, spin precession frequency blows up as the observer reaches the central object while for naked singularity it remains finite except at the ring singularity. Moreover, we study Lense-Thirring precession for a Kerr-like black hole and geodetic precession for Schwarzschild black hole in perfect fluid dark matter. To this end, we have calculated the Kepler frequency (KF), the vertical epicyclic frequency (VEF), and the nodal plane precession frequency (NPPF). Our results show that, the PFDM parameter $\alpha$ significantly affects those frequencies. This difference can be used by astrophysical observations in the near future to shed some light on the nature of dark matter.