Tidally-induced nonlinear resonances in EMRIs with an analogue model

Abstract

One of the important targets for the future space-based gravitational wave observatory LISA is extreme mass ratio inspirals (EMRIs), where long and accurate waveform modeling will be necessary for detection and characterization. Modeling EMRI dynamics requires accounting for effects such as the ones induced by an external tidal field, which can break integrability at resonances and cause significant dephasing. In this paper, we use a Newtonian analogue of a Kerr black hole to study the effect of an external tidal field on the dynamics and the gravitational waveform. We present a numerical framework that takes advantage of the integrability of the background system to evolve it with a symplectic splitting integrator and compute approximate gravitational waveforms to estimate the timescale over which the perturbation affects the dynamics. Comparing this timescale with the characteristic time under radiation reaction at resonance, we introduce a tool for quantifying the regime in which tidal effects might be included when modeling EMRI gravitational waves. As an application of this framework, we perform a detailed analysis of the dynamics at one resonance to show how different entry points into the resonance in phase-space can produce substantially different dynamics, and how one can estimate bounds for the parameter space where tidal effects may become dominant. Such bounds will scale as $\varepsilon \gtrsim C \, q$, where $\varepsilon$ measures the strength of the external tidal field, $q$ is the mass ratio, and $C$ is a number which depends on the resonance and the shape of the tide. We demonstrate how to estimate $C$ using our framework for the 2:3 radial to polar frequency resonance in our model system. This framework can serve as a proxy for proper modeling of the tidal perturbation in the fully relativistic case.

Resonant dynamics of extreme mass-ratio inspirals in a perturbed Kerr spacetime

“…The near-resonance orbits can be classified into three categories: orbits in the rotation regime (blue line), chaotic transitional orbits (orange line), and orbits in the libration regime (green horizontal line). Similar features in the rotation curve have been found in previous studies on various non-Kerr metrics (see, e.g., [38][39][40][41][42][43][44][45][46][47][48]). These features (including both the plateau and jumps in the rotation curve) were believed to be the signatures of chaos.…”

“…The limitation is that there is no chaos in this one-degree-of-freedom description, because the number of degrees of freedom is equal to the number of conserved quantity (the effective energy). In reality, we find the layers of chaos (as previous observed in various spacetimes [38][39][40]46]) lying between the rotational and libration orbits, thanks to the contribution from nonresonant terms.…”

“…In the resonant regime, i.e., kΩ r þ mΩ ϕ þ nΩ θ ≈ 0 for k; m; n ∈ Z, where Ω r;θ;ϕ are geodesic orbital frequencies in r; θ; ϕ directions, the KAM theorem no longer applies and orbital chaos can occur. In fact, signatures of chaotic orbits near resonances have been observed in studies of various modified Kerr spacetimes [38][39][40][41][42][43][44][45][46][47][48][49], e.g., in the vertical jumps of rotation numbers and volume-filling features in the phase space of the trajectory. A plateau in rotation number is sometimes observed, which should be associated with the resonant islands in the phase space.…”

“…II. This Hamiltonian governs the essential dynamics of the EMRI system within the resonant islands (commonly referred to as the "libration" regime for planetary systems), and it applies for generic metric perturbations h, including all the specific examples mentioned in previous studies [38][39][40][41][42][43][44][45][46][47][48][49]. The physical essence of this effective Hamiltonian-similar to the cases of mean motion resonances widely studied in planetary systems-is that there is a single resonant d.o.f.…”