Analytical computation of quasi-normal modes of slowly-rotating black-holes in dCS gravity

Using gravitational wave observations to search for deviations from general relativity in the strong-gravity regime has become an important research direction. Chern Simons (CS) gravity is one of the most frequently studied parity-violating models of strong gravity. It is known that the Kerr black-hole is not a solution for CS gravity. At the same time, the only rotating solution available in the literature for dynamical CS (dCS) gravity is the slow-rotating case most accurately known to quadratic order in spin. In this work, for the slow-rotating case (accurate to first order in spin), we derive the linear perturbation equations governing the metric and the dCS field accurate to linear order in spin and quadratic order in the CS coupling parameter ($\alpha$) and obtain the quasi-normal mode (QNM) frequencies. After confirming the recent results of Wagle et al. (2021), we find an additional contribution to the eigenfrequency correction at the leading perturbative order of $\alpha^2$. Unlike Wagle et al., we also find corrections to frequencies in the polar sector. We compute these extra corrections by evaluating the expectation values of the perturbative potential on unperturbed QNM wavefunctions along a contour deformed into the complex-$r$ plane. For $\alpha=0.1 M^2$, we obtain the ratio of the imaginary parts of the dCS correction to the GR correction in the first QNM frequency (in the polar sector) to be $0.263$ implying significant change. For the $(2,2)-$mode, the dCS corrections make the imaginary part of the first QNM of the fundamental mode less negative, thereby decreasing the decay rate. Our results, along with future gravitational wave observations, can be used to test for dCS gravity and further constrain the CS coupling parameters. [abridged]