Axial quasi-normal modes of slowly rotating black holes in dynamical Chern-Simons gravity to second-order in spin and coupling

Abstract

We compute the quasi-normal mode (QNM) frequencies of slowly rotating black holes in dynamical Chern-Simons (dCS) gravity, including corrections up to second order in the black hole's dimensionless spin parameter $χ= J/M^2$ and second order in the dCS coupling parameter ($α$). Due to the complexities of constructing a Newman-Penrose tetrad at this order, we employ a metric perturbation approach. We derive a system of coupled ordinary differential equations for the primary axial $l$-mode and the polar $l\pm 1$ modes, which is then solved numerically with appropriate ingoing and outgoing wave boundary conditions. Our numerical framework is validated in the General Relativistic limit against known Schwarzschild QNMs and highly accurate Kerr QNM results for $χ\leq 0.15$. For the fundamental $n=0, l=m=2$ axial mode, we present detailed numerical results illustrating the dependence of QNM frequencies on both $χ$ and $α$. We observe that while rotation generally increases the damping time, increasing the dCS coupling parameter significantly reduces the damping time of the axial mode. This finding contrasts with previous analytical work on polar modes, which suggested an increase in damping time due to dCS effects, highlighting a crucial parity-dependent difference in how dCS gravity impacts black hole ringdowns. Furthermore, we provide an analytical fitting formula for this mode. These results, incorporating coupled spin and dCS effects at second order, provide more accurate theoretical predictions for testing dCS gravity with gravitational wave observations of black hole ringdowns. The refined QNM calculations are particularly relevant for lower-mass black hole merger events, such as GW230529, where dCS corrections may be more prominent and their distinct damping signatures could be observable. [Abridged Version]