New template family for the detection of gravitational waves from comparable-mass black hole binaries

In order to improve the phasing of the comparable-mass waveform as we approach the last stable orbit for a system, various resummation methods have been used to improve the standard post-Newtonian waveforms. In this work we present a new family of templates for the detection of gravitational waves from the inspiral of two comparable-mass black hole binaries. These new adiabatic templates are based on reexpressing the derivative of the binding energy and the gravitational wave flux functions in terms of shifted Chebyshev polynomials. The Chebyshev polynomials are a useful tool in numerical methods as they display the fastest convergence of any of the orthogonal polynomials. In this case they are also particularly useful as they eliminate one of the features that plagues the post-Newtonian expansion. The Chebyshev binding energy now has information at all post-Newtonian orders, compared to the post-Newtonian templates which only have information at full integer orders. In this work, we compare both the post-Newtonian and Chebyshev templates against a fiducially exact waveform. This waveform is constructed from a hybrid method of using the test-mass results combined with the mass dependent parts of the post-Newtonian expansions for the binding energy and flux functions. Our results show that the Chebyshev templates achieve extremely high fitting factors at all post-Newtonian orders and provide excellent parameter extraction. We also show that this new template family has a faster Cauchy convergence, gives a better prediction of the position of the last stable orbit and in general recovers higher Signal-to-Noise ratios than the post-Newtonian templates.

Figure 1

The 7th order shifted Chebyshev polynomials for a $(10,10)M_{\odot }$ BH-BH binary in the domain $v\in [0.23,0.408]$. We can see that the polynomials have $n+1$ alternating maximum and minimum values of $\pm 1$ and $n$ zeros across the domain.

Figure 2

A comparison of the convergence for the binding energy derivative coefficients of the PN approximation (left) and the shifted Chebyshev approximation (right) for the four BH-BH systems with masses (5, 5), (10, 10), (20, 10) and $(20,5)M_{\odot }$. We can see that while the values of the PN coefficients oscillate wildly, the Chebyshev coefficients get smaller as we go to higher orders of approximation. We have plotted two systems with $\eta =0.25$ to emphasize the fact that the Chebyshev coefficients ${\mathcal{E}}_k={\mathcal{E}}_k(m)$ are a function of the total mass of the system.

Figure 3

Comparison of the relative errors for the PN (left) and Chebyshev (right) energy derivative functions against a fiducial exact energy function for four BH-BH systems with masses (5, 5), (10, 10), (20, 10) and $(20,5)M_{\odot }$. Each dip in the approximation error curves correspond to a crossing of the exact energy function by the approximated flux. The initial velocity in each case is defined by a lower frequency cutoff of 40 Hz.

Figure 4

A comparison of the convergence for the flux coefficients of the PN approximation (left) and the shifted Chebyshev approximation (right) for the four BH-BH systems with masses (5, 5), (10, 10), (20, 10) and $(20,5)M_{\odot }$. We can see that the values of the PN coefficients again oscillate wildly and display an overall trend of growth. While not as smooth as in the case of the binding energy, the Chebyshev coefficients again get smaller as we go to higher orders of approximation.

Figure 5

Comparison of the relative error for the PN (left) and Chebyshev (right) GW flux functions against a fiducial exact flux function for four BH-BH systems with masses (5, 5), (10, 10), (20, 10) and $(20,5)M_{\odot }$. Each dip in the approximation error curve corresponds to a crossing of the exact energy function by the approximated flux. The initial velocity in each case is defined by a lower frequency cutoff of 40 Hz.

Figure 6

The percentage errors in total mass $m$, reduced mass ratio $\eta$ and fitting factors for three equal mass systems of (5, 5), (10, 10) and $(20,20)M_{\odot }$ for both PN and Chebyshev waveforms when compared to the fiducially exact hybrid waveform.

Figure 7

The percentage errors in total mass $m$, reduced mass ratio $\eta$ and fitting factors for three unequal mass systems of (10, 5), (20, 10) and $(20,5)M_{\odot }$ for both PN and Chebyshev waveforms when compared to the fiducially exact hybrid waveform.

Figure 8

Comparison of the position of the LSO for the PN and Chebyshev energy functions against the minimum of the fiducial exact energy function for the four BH-BH systems with masses (5, 5), (10, 10), (20, 10) and $(20,5)M_{\odot }$. The circles denote the LSO for the exact fiducial energy, the squares represent the PN approximation and the triangles represent the Chebyshev approximation. We can see that the Chebyshev approximation is more convergent and gives a better prediction of the location of the LSO at virtually all PN orders.

Figure 9

Comparison of the recovered SNRs for both PN and Chebyshev templates using the fiducially exact signal for equal mass systems. The sources are set at a common distance of 100 Mpc.