Improved resummation of post-Newtonian multipolar waveforms from circularized compact binaries

We improve and generalize a resummation method of post-Newtonian multipolar waveforms from circular (nonspinning) compact binaries introduced in Refs. [1,2]. One of the characteristic features of this resummation method is to replace the usual additive decomposition of the standard post-Newtonian approach by a multiplicative decomposition of the complex multipolar waveform h_m into several (physically motivated) factors: (i) the Newtonian waveform, (ii) a relativistic correction coming from an ``effective source,'' (iii) leading-order tail effects linked to propagation on a Schwarzschild background, (iv) a residual tail dephasing, and (v) residual relativistic amplitude corrections f_m. We explore here a new route for resumming f_m based on replacing it by its -th root: ρ_m=f_m^1/. In the extreme-mass-ratio case, this resummation procedure results in a much better agreement between analytical and numerical waveforms than when using standard post-Newtonian approximants. We then show that our best approximants behave in a robust and continuous manner as we deform them by increasing the symmetric mass ratio ν≡m₁m₂/(m₁+m₂)² from 0 (extreme-mass-ratio case) to $1/4$ (equal-mass case). The present paper also completes our knowledge of the first post-Newtonian corrections to multipole moments by computing ready-to-use explicit expressions for the first post-Newtonian contributions to the odd-parity (current) multipoles.