Two timescale analysis of extreme mass ratio inspirals in Kerr. I. Orbital Motion

TL;DR

The paper develops a rigorous two-timescale expansion for extreme-mass-ratio inspirals in Kerr spacetime, casting the orbital dynamics in generalized action-angle variables and showing that the leading adiabatic motion is governed by the orbit-averaged dissipative self-force. It extends the framework to post-adiabatic corrections by including oscillatory and conservative pieces, and provides explicit results for single and multi-degree-of-freedom systems under a no-resonance assumption. A sketch of the two-timescale expansion of Einstein’s equations yields analytic adiabatic waveforms, and the authors discuss the implications for waveform generation and gravitational-wave data analysis, including the radiative approximation and its limitations. The work establishes a consistent, unique perturbative scheme that separates fast orbital dynamics from slow inspiral, laying groundwork for accurate EMRI templates and informing future treatments of resonances and full waveform modeling.

Abstract

Inspirals of stellar mass compact objects into massive black holes are an important source for future gravitational wave detectors such as Advanced LIGO and LISA. Detection of these sources and extracting information from the signal relies on accurate theoretical models of the binary dynamics. We cast the equations describing binary inspiral in the extreme mass ratio limit in terms of action angle variables, and derive properties of general solutions using a two-timescale expansion. This provides a rigorous derivation of the prescription for computing the leading order orbital motion. As shown by Mino, this leading order or adiabatic motion requires only knowledge of the orbit-averaged, dissipative piece of the self force. The two timescale method also gives a framework for calculating the post-adiabatic corrections. For circular and for equatorial orbits, the leading order corrections are suppressed by one power of the mass ratio, and give rise to phase errors of order unity over a complete inspiral through the relativistic regime. These post-1-adiabatic corrections are generated by the fluctuating piece of the dissipative, first order self force, by the conservative piece of the first order self force, and by the orbit-averaged, dissipative piece of the second order self force. We also sketch a two-timescale expansion of the Einstein equation, and deduce an analytic formula for the leading order, adiabatic gravitational waveforms generated by an inspiral.

Figures (4)

• Figure 1: The parameter space of inspiralling compact binaries in general relativity, in terms of the inverse mass ratio $M/\mu = 1/\varepsilon$ and the orbital radius $r$, showing the different regimes and the computational techniques necessary in each regime. Individual binaries evolve downwards in the diagram (green dashed arrows).
• Figure 2: The exact numerical solution of the system of equations (\ref{['examplesystem']}). After a time $\sim 1/\varepsilon$, the action variable $J$ is $O(1)$, while the angle variable $q$ is $O(1/\varepsilon)$.
• Figure 3: Upper panels: The difference between the solution of the exact dynamical system (\ref{['examplesystem']}) and the adiabatic approximation given by Eqs. (\ref{['adiabaticexample']}) and (\ref{['adiabaticexample1']}). For the action variable $J$, this difference is $O(\varepsilon)$, while for the angle variable $q$, this difference is $O(1)$, as expected. Lower panels: The difference between the exact solution and the post-1-adiabatic approximation given by Eqs. (\ref{['adiabaticexample']}), (\ref{['p1adiabaticexample']}) and (\ref{['p1xexamplesoln']}). Again the magnitudes of these errors are as expected: $O(\varepsilon^2)$ for $J$ and $O(\varepsilon)$ for $q$.
• Figure 4: The maximum orbital phase error in cycles, $\delta N = \delta \phi/(2 \pi)$, incurred in the radiative approximation during the last year of inspiral, as a function of the mass $M_6$ of the central black hole in units of $10^6 M_\odot$, the mass $\mu_{10}$ of the small object in units of $10 \, M_\odot$, and the eccentricity $e$ of the system at the start of the final year of inspiral. The exact and radiative inspirals are chosen to line up at some time $t_{\rm m}$ during the final year, and the value of $t_{\rm m}$ is chosen to minimize the phase error. The initial data at time $t_{\rm m}$ for the radiative evolution is slightly different to that used for the exact evolution in order that the secular pieces of the two evolutions initially coincide; this is the "time-averaged" initial data prescription of Pound and Poisson Pound:2007th. All evolutions are computed using the hybrid equations of motion of Kidder, Will and Wiseman PhysRevD.47.3281 in the osculating-element form given by Pound and Poisson.