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 the shadow radius of black holes (BHs) and then explore the effect of dark matter on the QNMs of massless scalar field and electromagnetic field perturbations in a BH spacetime surrounded by perfect fluid dark matter (BHPFDM). Using the Wentzel--Kramers--Brillouin (WKB) approximation, we show that the quasinormal mode spectra of BHPFDM deviate from those of Schwarzschild black hole due to the presence of the perfect fluid dark matter (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 decay more rapidly compared to the Schwarzschild vacuum BH. We point out that there exists a reflecting point k₀ corresponding to maximal values for the real part of QNM frequencies. Namely, as the PFDM parameter k increases in the interval k<k₀, the QNM frequencies increases and reach their maximum values at k=k₀. Finally, we show that k₀ 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.