A new and improved template for the detection of gravitational waves from compact binary systems using Chebyshev polynomials

TL;DR

This paper introduces a Chebyshev polynomial-based template for gravitational wave detection from compact binaries, demonstrating improved accuracy and convergence over traditional methods in the test-mass regime.

Contribution

The study develops a Chebyshev polynomial approximation for the gravitational wave flux, outperforming post-Newtonian and Pade methods in accuracy and convergence.

Findings

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

Templates achieve fitting factors >0.99 and mass estimation errors <0.1%.

Chebyshev templates outperform PN and Pade at lower orders.

Abstract

We introduce a new template for the detection of gravitational waves from compact binary systems which is based on Chebyshev polynomials of the first kind. As well as having excellent convergence properties, these polynomials are also very closely related to the elusive minimax polynomial. In this study we have limited ourselves to the test-mass regime, where we model a test particle in a circular equatorial orbit around a Schwarzschild black hole. Our objective is to model the numerical gravitational wave flux function, 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 better model than either post-Newtonian or Pade based methods. A graphical examination of the new flux function shows that it gives an excellent fit to the numerical flux, but more importantly we find that at the last stable orbit the error 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 test-mass case, the new Chebyshev template is superior to both PN and Pade approximant templates, especially at lower orders of approximation.