Probing the black holes in a dark matter halo of M87 using gravitational wave echoes

Abstract Variations at the event horizon structure of a black hole will emit the signals of the gravitational wave echoes associated with the ringdown of the binary black hole merger. In this work, combining mass model of M87 and Einasto profile for dark matter halo, we construct one formal solution of black holes in dark matter halo, and this solution includes the regular black hole for geometric parameter $$a &lt; 2M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>2</mml:mn> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> and the wormhole for $$a \ge 2M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> . Then, we test this solution under axial gravitational perturbation and calculate their quasinormal modes (QNM). Our results show that when geometric parameter $$a&gt;2M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> , a series of gravitational wave echoes appear after the QNM, and one distinctive feature of gravitational wave echoes is the double barrier. Besides, we also study the impacts of shape parameters $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> of Einasto profile both on the QNM and gravitational wave echoes, and extract their frequencies using WKB method and Prony method. In principle, with the increasing of shape parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , the frequencies of the QNM and gravitational wave echoes between the different shape parameters must be different. But now there is zero difference of the frequencies between shape parameter $$\alpha =0.18$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.18</mml:mn> </mml:mrow> </mml:math> and $$\alpha =0.20$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.20</mml:mn> </mml:mrow> </mml:math> . Zero difference is not allowed, so we give an upper limit for shape parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , which is approximately $$\alpha &lt;0.18$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>0.18</mml:mn> </mml:mrow> </mml:math> . These signals of gravitational wave echoes in a dark matter halo may be detected in the near future, and these characteristics of gravitational wave echoes may serve as local measurements of dark matter.