Spicing up the recipe for echoes from exotic compact objects: orbital differences and corrections in rotating backgrounds

Recently it has been argued that near-horizon modifications of the standard (classical) black hole spacetime could lead to observable alterations of the gravitational waveform generated by a binary black hole coalescence. Such modifications can be inspired by quantum gravity considerations, resulting in speculative horizonless exotic compact objects (ECOs) with no singularities, which may be an alternative to the classical black hole solution. A largely model-independent description of these objects proposed in the literature relies on the introduction of a partially reflective wall at some small distance away from the "would-be" horizon. The inspiral-merger-ringdown of a pair of such objects would be subject to possibly detectable deviations from the black hole case due to matter effects. In particular, the ringdown phase would be modified by the late emergence of so-called "echoes" in the waveform, but most studies so far have considered spherically symmetric backgrounds. We use an in-falling scalar charge as an initial perturbation to simulate the excitation of the echoes of a rotating ECO and we explore both the co-rotating and counter-rotating cases, which provide distinct signals. In particular, rotation breaks the symmetry between positive and negative frequencies and introduces a subdominant frequency contribution in each echo, which we examine here for the first time. Our results follow consistently from the solution of the Teukolsky equation using the MST method developed by Mano, Suzuki and Takasugi, and the construction of the complex Green's function integrated over different particle geodesics.