Ineffectiveness of Padé resummation techniques in post-Newtonian approximations

Abstract

We test the resummation techniques used in developing Padé and effective one body (EOB) waveforms for gravitational wave detection. Convergence tests show that Padé 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 Padé transformations should help in estimating parameters better in data analysis. Moreover, adding a pole to the flux seems unnecessary in the construction of these Padé-approximated flux formulas. Padé 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 Padé waveforms of Damour et al. in the equal mass, nonspinning quasicircular case. The comparison suggests that the Padé 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 Taylor-expanded 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. The 4PN parameter incorporates higher order effects of the radiation reaction. 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 Padé transformation in its construction, but this is the only place Padé transformations seem useful.