Spherical inspirals of spinning bodies into Kerr black holes

TL;DR

This paper develops a fully relativistic framework for flux-driven, nearly spherical EMRI orbits with a spinning secondary in Kerr spacetime, keeping terms linear in the secondary spin and incorporating 1PA phasing. It combines analytic solutions for spinning-particle motion with Teukolsky-based GW fluxes, enabling frequency-domain waveforms via an extended stationary-phase approximation and a two-timescale evolution of orbital elements. The authors demonstrate that spherical orbits remain spherical on average under radiation reaction, derive the corresponding orbital-evolution equations, and quantify the impact of the secondary spin on the GW signal through mismatches, showing potential detectability with LISA in many configurations. This work lays a foundation for including secondary-spin effects in EMRI templates and points to future extensions to fully generic bound orbits and comprehensive Bayesian analyses.

Abstract

Extreme mass-ratio inspirals (EMRIs), consisting of a stellar-mass compact object spiraling into a massive black hole, are key sources for future space-based gravitational wave observatories such as LISA. Accurate modeling of these systems requires incorporating the spin effects of both the primary and secondary bodies, particularly for waveforms at the precision required for LISA detection and astrophysical parameter extraction. In this work, we develop a framework for modeling flux-driven spherical inspirals (orbits of approximately constant Boyer-Lindquist radius) of a spinning secondary into a Kerr black hole. We leverage recently found solutions for the motion of spinning test particles and compute the associated gravitational wave fluxes to linear order in the secondary spin. Next, we show that spherical orbits remain spherical under radiation reaction at linear order in spin, and derive the evolution of the orbital parameters throughout the inspiral. We implement a numerical scheme for waveform generation in the frequency domain and assess the impact of the secondary spin on the gravitational wave signal. In contrast to quasi-circular inspirals, we find that neglecting the secondary spin in spherical inspirals induces large mismatches in the waveforms that will plausibly be detectable by LISA.

Figures (12)

• Figure 1: Projection of the near-spherical trajectories into the $r-z$ plane for different values of $s_\parallel$ with fixed turning points. The orbital parameters are $a = 0.95 M$, $p=8M$, and $x=0.5$. The secondary spin causes a slight bending of the orbit away from a trajectory of constant radius, which led us to call this class of trajectories only "nearly spherical". (The value of $s_\parallel$ is unphysically large to make the cases distinguishable; typical EMRI parameters would show much smaller effects.)
• Figure 2: Radius of the innermost spherical geodesics $r_{\rm ISSOg}$ and the secondary spin correction $\delta r_{\rm ISSO}$ as a function of the inclination parameter $x$ for several values of primary black hole spin.
• Figure 3: Dependence of the linear part of the radial action with fixed turning points $\eval{\delta J_r}_{p,e,x}$ on eccentricity $e$ for different orbital parameters.
• Figure 4: Evolution of the adiabatic $p_\text{g}$ and $x_\text{g}$ for $a=0.95M$ (solid) and $0.525M$ (dashed) and final $x_\text{f} = 0.38$, $0.66$, and $0.94$. The black lines show the respective ISSO positions.
• Figure 5: Evolution of the linear corrections $\delta p$ (top) and $\delta x$ (bottom) for inspirals with parameters $M=10^6 M_\odot$, $q=10^{-4}$. The inclination shifts stay roughly constant up to the ISSO, while the radius shifts dynamically evolve throughout the inspirals.