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 $a\ensuremath{\equiv}J/{M}^{2}$ (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 ${v}^{8}$ 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 $v/{v}_{\mathrm{ISCO}}$, where ${v}_{\mathrm{ISCO}}$ 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 ${v}^{6}$. Independent of $a$, the inclusion of angular multipoles up to and including $\ensuremath{\ell}=5$ 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.