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 a black hole if a <= a(c) and naked singularity if a > a(c). Also, for -2 <= alpha < 2/3, the size of the black hole horizons decreases, whereas for 2/3 < alpha it increases. We also study the spin precession frequency of a test gyroscope attached to a stationary observer to differentiate a black hole from naked singularity in PFDM. 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 a Schwarzschild black hole in PFDM. 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.