Quasinormal Modes of Black Holes Surrounded by Dark Matter and Their Connection with the Shadow Radius

The purpose of this article is twofold. First, we highlight a simple connection between the real part of quasinormal modes (QNMs) in the eikonal limit and shadow radius of BHs and then explore the effect of dark matter on the QNMs of massless scalar field and electromagnetic field perturbations in a black hole (BH) spacetime surrounded by perfect fluid dark matter (BHPFDM). Using the WKB approximation we show that the quasinormal mode spectra of BHPFDM deviate from those of Schwarzschild black hole due to the presence of the PFDM encoded by the parameter $k$. Moreover it is shown that for any $k>0$, the real part and the absolute value of the imaginary part of QNM frequencies increases and this means that the field perturbations in the presence of PFDM decays more rapidly compared to Schwarzschild vacuum BH. We point out that there exists a reflecting point $k_0$ corresponding to maximal values for the real part of QNM frequencies. Namely, as the PFDM parameter $k$ increases in the interval $k<k_0$, the QNM frequencies increases and reach their maximum values at $k=k_0$. Finally we show that $k_0$ is also a reflecting point for the shadow radius while this conclusion can be deduced directly from the inverse relation between the real part of QNMs and the shadow radius.