Ineffectiveness of Pad\'e resummation techniques in post-Newtonian approximations

TL;DR

This paper critically evaluates Padé resummation techniques in gravitational wave modeling, finding they do not improve convergence or waveform accuracy over standard methods, but may still assist in formulating fitting functions.

Contribution

The study demonstrates the limited effectiveness of Padé approximants in post-Newtonian waveform modeling and proposes a simple EOB model with minimal phase difference to numerical data.

Findings

Padé approximants do not accelerate convergence of gravitational wave flux.

Padé waveforms do not outperform standard Taylor waveforms in matching numerical data.

A simple EOB model with a fitted parameter achieves a phase difference of only 0.002 radians.

Abstract

We test the resummation techniques used in developing Pad\'e and Effective One Body (EOB) waveforms for gravitational wave detection. Convergence tests show that Pad\'e approximants of the gravitational wave energy flux do not accelerate the convergence of the standard Taylor approximants even in the test mass limit, and there is no reason why Pad\'e transformations should help in estimating parameters better in data analysis. Moreover, adding a pole to the flux seems unnecessary in the construction of these Pad\'e-approximated flux formulas. Pad\'e approximants may be useful in suggesting the form of fitting formulas. We compare a 15-orbit numerical waveform of the Caltech-Cornell group to the suggested Pad\'e waveforms of Damour et al. in the equal mass, nonspinning quasi-circular case. The comparison suggests that the Pad\'e waveforms do not agree better with the numerical waveform than the standard Taylor based waveforms. Based on this result, we design a simple EOB model by modifiying the ET EOB model of Buonanno et al., using the Taylor series of the flux with an unknown parameter at the fourth post-Newtonian order that we fit for. This simple EOB model generates a waveform having a phase difference of only 0.002 radians with the numerical waveform, much smaller than 0.04 radians the phase uncertainty in the numerical data itself. An EOB Hamiltonian can make use of a Pad\'e transformation in its construction, but this is the only place Pad\'e transformations seem useful.