Quasi-normal modes of slowly-rotating Johannsen black holes

Abstract The detection of gravitational waves with ground-based laser interferometers has opened a new window to test and constrain General Relativity (GR) in the strong, dynamical, and non-linear regime. In this paper, we follow an agnostic approach and we study the quasi-normal modes of gravitational perturbations of Johannsen black holes under the assumptions of the validity of the Einstein Equations and of low values of the black hole spin parameter and deformation parameters. We find that the deformation parameter $$\alpha _{13}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mn>13</mml:mn> </mml:msub> </mml:math> has a stronger impact on the quasi-normal modes than the other leading order deformation parameters ( $$\alpha _{22}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mn>22</mml:mn> </mml:msub> </mml:math> , $$\alpha _{52}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mn>52</mml:mn> </mml:msub> </mml:math> , and $$\epsilon _{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϵ</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> ). We derive a fitting formula for the fundamental modes with $$l=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and $$l=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> for the deformation parameter $$\alpha _{13}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mn>13</mml:mn> </mml:msub> </mml:math> valid in the slow rotation approximation ( $$a_* &lt; 0.4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mmultiscripts> <mml:mi>a</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> <mml:mo>&lt;</mml:mo> <mml:mn>0.4</mml:mn> </mml:mrow> </mml:math> ). Finally, we constrain $$\alpha _{13}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mn>13</mml:mn> </mml:msub> </mml:math> from the event GW170104; within our analysis, we find that the data of GW170104 are consistent with the predictions of GR.