Accuracy of the Post-Newtonian Approximation: Optimal Asymptotic Expansion for Quasi-Circular, Extreme-Mass Ratio Inspirals

TL;DR

This paper assesses the accuracy and validity range of the post-Newtonian approximation for extreme-mass ratio inspirals, finding optimal accuracy at 3PN order and suggesting divergence beyond that due to logarithmic terms.

Contribution

It introduces an asymptotic analysis approach to determine the validity region of the PN approximation for black hole inspirals, highlighting the importance of higher multipoles and the 3PN order.

Findings

Optimal validity at 3PN order (O(1/c^6))

Higher multipoles are essential for accuracy

PN series likely diverges beyond 3PN due to logarithmic terms

Abstract

We study the accuracy of the post-Newtonian (PN) approximation and its formal region of validity, by investigating its optimal asymptotic expansion for the quasi-circular, adiabatic inspiral of a point particle into a Schwarzschild black hole. By comparing the PN expansion of the energy flux to numerical calculations in the perturbative Teukolsky formalism, we show that (i) the inclusion of higher multipoles is necessary to establish the accuracy of high-order PN terms, and (ii) the region of validity of PN theory is largest at relative O(1/c^6) (3PN order). The latter result suggests that the series diverges beyond 3PN order, at least in the extreme mass-ratio limit, probably due to the appearance of logarithmic terms in the energy flux. The study presented here is a first formal attempt to determine the region of validity of the PN approximation using asymptotic analysis. Therefore, it should serve as a template to perform similar studies on other systems, such as comparable-mass quasi-circular inspirals computed by high-accuracy numerical relativistic simulations.