Gravitational waves from binary systems in circular orbits: Does the post-Newtonian expansion converge?

TL;DR

This paper investigates the convergence of the post-Newtonian series for gravitational wave energy flux from binary systems in circular orbits, finding conditions under which the truncated series converges, relevant for modeling inspirals.

Contribution

The study provides explicit convergence criteria for a simplified model of gravitational radiation, clarifying the role of Newtonian-like moments in the series convergence.

Findings

Series converges for v/c < 2/e with a small mass ratio

Series converges for v/c < 4/e with equal masses

Convergence depends on the truncated model, not the full series

Abstract

Gravitational radiation can be expressed in terms of an infinite series of radiative, symmetric trace-free (STF) multipole moments which can be connected to the behavior of the source. We consider a truncated model for gravitational radiation from binary systems in which each STF mass and current moment of order l is given by the lowest-order, Newtonian-like l-pole moment of the orbiting masses; we neglect post-Newtonian corrections to each STF moment. Specializing to orbits which are circular (apart from the radiation-induced inspiral), we find an explicit infinite series for the energy flux in powers of $v/c$, where v is the orbital velocity. We show that the series converges for all values $v/c < 2/e$ when one mass is much smaller than the other, and $v/c < 4/e$ for equal masses,where e is the base of natural logarithms. These values include all physically relevant values for compact binary inspiral. This convergence cannot indicate whether or not the full series (obtained from the exact moments) will converge. But if the full series does not converge, our analysis shows that this failure to converge does not originate from summing over the Newtonian part of the multipole moments.