Dynamics of black hole in dark matter halo: Quasinormal modes

In this paper, we explore the scalar, electromagnetic, and gravitational perturbations of a Schwarzschild black hole embedded within a Dehnen-(1, 4, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>γ</a:mi></a:math>) dark matter halo. We focus on three distinct values of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>γ</c:mi></c:math>, each yielding a unique central behavior for the dark matter distribution: <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>γ</e:mi><e:mo>=</e:mo><e:mn>1</e:mn></e:math>, where the central density vanishes; <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>γ</g:mi><g:mo>=</g:mo><g:mn>0</g:mn></g:math>, where it remains finite; and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>γ</i:mi><i:mo>=</i:mo><i:mn>1</i:mn></i:math>, where it diverges. Employing higher-order Wentzel-Kramers-Brillouin and asymptotic iteration methods, we calculate the characteristic oscillation frequencies, namely, the quasinormal modes and present their temporal evolution using a time-domain approach. Our findings indicate that dark matter amplifies the gravitational pull of the black hole. While it reduces the actual oscillation frequencies of the perturbations, its impact on the damping rates of the fundamental modes is negligible. However, in the nonfundamental modes, both the real and imaginary parts of the quasinormal frequencies exhibit a pronounced decay.