Accuracy of the post-Newtonian approximation. II. Optimal asymptotic expansion of the energy flux for quasicircular, extreme mass-ratio inspirals into a Kerr black hole

TL;DR

This paper examines how black hole spin affects the accuracy of the post-Newtonian approximation for energy flux in extreme mass-ratio inspirals, finding weak spin dependence and emphasizing the importance of including multipoles up to l=5 for accuracy.

Contribution

It generalizes previous work by analyzing the validity region of the post-Newtonian expansion for spinning black holes and explores the behavior of the error function with respect to spin and order.

Findings

Region of validity weakly depends on black hole spin a.

Including multipoles up to l=5 is necessary for accurate flux calculations.

No clear transition from convergence to divergence at order v^6 observed.

Abstract

We study the effect of black hole spin on the accuracy of the post-Newtonian approximation. We focus on the gravitational energy flux for the quasicircular, equatorial, extreme mass-ratio inspiral of a compact object into a Kerr black hole of mass M and spin J. For a given dimensionless spin a=J/M^2 (in geometrical units), the energy flux depends only on the orbital velocity v or (equivalently) on the Boyer-Lindquist orbital radius r. We investigate the formal region of validity of the Taylor post-Newtonian expansion of the energy flux (which is known up to order v^8 beyond the quadrupole formula), generalizing previous work by two of us. The "error function" used to determine the region of validity of the post-Newtonian expansion can have two qualitatively different kinds of behavior, and we deal with these two cases separately. We find that, at any fixed post-Newtonian order, the edge of the region of validity (as measured by v/v_{ISCO}, where v_{ISCO} is the orbital velocity at the innermost stable circular orbit) is only weakly dependent on a. Unlike in the nonspinning case, the lack of sufficiently high order terms does not allow us to determine if there is a convergent to divergent transition at order v^6. Independently of a, the inclusion of angular multipoles up to and including l=5 in the numerical flux is necessary to achieve the level of accuracy of the best-known (N=8) PN expansion of the energy flux.