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

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[superscript 2] (in geometrical units G=c=1), 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[superscript 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[subscript ISCO], where v[subscript 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[superscript 6]. Independent 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) post-Newtonian expansion of the energy flux.