Modelling the post-Newtonian test-mass gravitational wave flux function for compact binary systems using Chebyshev polynomials

TL;DR

This paper introduces a Chebyshev polynomial-based method for modeling gravitational wave flux functions in compact binary systems, achieving faster convergence and higher accuracy than traditional methods, especially at lower approximation orders.

Contribution

The paper presents a novel Chebyshev polynomial approximation for gravitational wave flux, improving convergence and accuracy over post-Newtonian and Padé methods in intermediate mass ratio inspirals.

Findings

Chebyshev approximation reduces flux error to <1.8% at last stable orbit.

Templates using Chebyshev approximation achieve >0.99 fitting factor.

Chebyshev templates outperform PN and Padé templates at lower orders.

Abstract

We introduce a new method for modelling the gravitational wave flux function of a test-mass particle inspiralling into an intermediate mass Schwarzschild black hole which is based on Chebyshev polynomials of the first kind. It is believed that these Intermediate Mass Ratio Inspiral events (IMRI) are expected to be seen in both the ground and space based detectors. Starting with the post-Newtonian expansion from Black Hole Perturbation Theory, we introduce a new Chebyshev approximation to the flux function, which due to a process called Chebyshev economization gives a model with faster convergence than either post-Newtonian or Pad\'e based methods. As well as having excellent convergence properties, these polynomials are also very closely related to the elusive minimax polynomial. We find that at the last stable orbit, the error between the Chebyshev approximation and a numerically calculated flux is reduced, $< 1.8%$, at all orders of approximation. We also find that the templates constructed using the Chebyshev approximation give better fitting factors, in general $> 0.99$, and smaller errors, $< 1/10%$, in the estimation of the Chirp mass when compared to a fiducial exact waveform, constructed using the numerical flux and the exact expression for the orbital energy function, again at all orders of approximation. We also show that in the intermediate test-mass case, the new Chebyshev template is superior to both PN and Pad\'e approximant templates, especially at lower orders of approximation.