Dynamical perturbations around an extreme mass ratio inspiral near resonance

Abstract

Extreme mass ratio inspirals (EMRIs) -systems with a compact object orbiting a much more massive (e.g., galactic center) black hole -are of interest both as a new probe of the environments of galactic nuclei, and their waveforms are a precision test of the Kerr metric. This work focuses on the effects of an external perturbation due to a third body around an EMRI system. This perturbation will affect the orbit most significantly when the inner body crosses a resonance with the outer body, and result in a change of the conserved quantities (energy, angular momentum, and Carter constant) or equivalently of the actions, which results in a subsequent phase shift of the waveform that builds up over time. We present a general method for calculating the changes in action during a resonance crossing, valid for generic orbits in the Kerr spacetime. We show that these changes are related to the gravitational waveforms emitted by the two bodies (quantified by the amplitudes of the Weyl scalar ψ4 at the horizon and at ∞) at the frequency corresponding to the resonance. This allows us to compute changes in the action variables for each body, without directly computing the explicit metric perturbations, and therefore we can carry out the computation by calling an existing black hole perturbation theory code. We show that our calculation can probe resonant interactions in both the static and dynamical limit. We plan to use this technique for future investigations of third-body effects in EMRIs and their potential impact on waveforms for LISA.

Resonant jumps induced by stationary tidal perturbation: a two-for-one deal

“…where we have defined n a ≡ {mΩ, n, k, m}. As we shall immediately see below, the above relation explains why the jumps of 'constants of motion' vanish when k + m = odd (see also the previous discussions [17,18]). Under the parity transformation ⃗ x → −⃗ x, the tidal field remains invariant.…”

“…It may, however, be necessary to evaluate a broader range of resonances in the future. Recently, a simpler method of computing the resonant jump for angular momentum was proposed [18]. However, this method does not directly apply to the jump in the Carter constant.…”