Suitability of hybrid gravitational waveforms for unequal-mass binaries

This article studies sufficient accuracy criteria of hybrid post-Newtonian (PN) and numerical relativity (NR) waveforms for parameter estimation of strong binary black-hole sources in second-generation ground-based gravitational-wave detectors. We investigate equal-mass nonspinning binaries with a new 33-orbit NR waveform, as well as unequal-mass binaries with mass ratios 2, 3, 4 and 6. For equal masses, the 33-orbit NR waveform allows us to recover previous results and to extend the analysis toward matching at lower frequencies. For unequal masses, the errors between different PN approximants increase with mass ratio. Thus, at 3.5 PN, hybrids for higher-mass-ratio systems would require NR waveforms with many more gravitational-wave cycles to guarantee no adverse impact on parameter estimation. Furthermore, we investigate the potential improvement in hybrid waveforms that can be expected from fourth-order post-Newtonian waveforms and find that knowledge of this fourth post-Newtonian order would significantly improve the accuracy of hybrid waveforms.

Figure 1

The trajectories of the black holes for the 33-orbit numerical simulation. The blue curve shows the trajectory of one black hole and the orange curve shows the trajectory of the second black hole. Also shown are the individual apparent horizons at $t=0$ and at the time when the common apparent horizon first forms, as well as the common apparent horizon.

Figure 2

Some of the waveforms used in this analysis. Shown is the real part of the (2,2) mode of the equal-mass, nonspinning waveform with 15 orbits [19, 30], the new 33-orbit simulation, and the waveform for a binary with mass ratio $q=6$ [34].

Figure 3

Phase error between numerical and post-Newtonian waveforms. A TaylorT3 $3.5/2.5\mathrm{PN}$ waveform is matched to NR at five different GW frequencies ${\omega }_m$, and the phase differences between PN and NR are plotted. The solid lines show the results obtained using the new 33-orbit numerical waveform, with filled diamonds indicating the location of the matching point. The thick dashed lines starting 16 orbits before merger represent the results achieved in Ref. 19.

Figure 4

Convergence and consistency tests between the 15- and 33- orbit NR waveforms for $q=1$. The dashed lines show truncation error of either waveform, the solid line shows the difference between 15- and 33-orbit waveforms. For all three curves, hybrids were constructed with the same Taylor approximant at the same matching frequency. $\Vert \delta h\Vert /\Vert h\Vert$ was then evaluated as a function of mass $M$ using the advanced LIGO noise curve.

Figure 5

Error of hybrid waveforms as a function of matching frequency for mass ratio $q=1$. The solid lines show $\Vert \delta h\Vert /\Vert h\Vert$ for hybrids constructed with different PN approximants and with the new 33-orbit NR waveform. The dashed lines show the previous results which use a 15-orbit NR waveform. These errors are all for binary black-hole systems with a total mass of $20M_{\odot }$.

Figure 6

The relationship between the number of orbits before merger and the gravitational wave frequency for mass ratios $q=1$ and $q=6$.

Figure 7

Accumulated phase difference between numerical and post-Newtonian waveforms for Taylor T1, T2, T3, and T4. PN and NR are matched at GW frequency $M{\omega }_m=0.1$, phase differences are then computed eight GW cycles earlier (top panel) or at GW frequency $M\omega =0.05$ (bottom panel). The Taylor T3 waveform used in this comparison is 3.0 PN order in phase.

Figure 8

Error of hybrid waveforms as a function of matching frequency for mass ratio $q=6$. Plotted is the value of $\Vert \delta h\Vert /\Vert h\Vert$ at 10 and $40M_{\odot }$ as a function of ${\omega }_m$ for a binary black-hole system with a mass ratio $q=6$. The data from Fig. 5 are plotted in grey in the background for reference.

Figure 9

Error of hybrid waveforms as a function of mass ratio $q$. PN and NR is matched at frequency $M{\omega }_m=0.046$ (top panel) or 12 orbits before merger (bottom panel). Pairwise differences between hybrids in either category are computed at total mass $M=20M_{\odot }$.

Figure 10

The magnitude of the PN coefficients for Taylor T4 as a function of PN order. Shown are different mass ratios and the test-mass limit. The straight black line shows our assumed coefficients for estimating the accuracy of 4 and 4.5 PN hybrids.

Figure 11

Estimated error of hybrid waveforms if PN up to the specified order were known. Shown is the equal-mass case. The line 3.5 PN represents the error in the presently available PN waveforms vs an estimated 4 PN term; see the text for details. The grey lines show the more exhaustive analysis from Fig. 5, which is consistent with the 3.5 PN line obtained with our alternative estimation procedure.

Figure 12

Estimated error of hybrid waveforms if PN up to the specified order were known. Shown is the mass ratio $q=6$ case. The line 3.5 PN represents the error in the presently available PN waveforms. The grey lines show the more exhaustive analysis from Fig. 8 but at a total mass of $20M_{\odot }$. Compare to Fig. 11.

Figure 13

Estimated reduction in error of hybrid waveforms constructed using 4 PN relative to those constructed with 3.5 PN. This reduction is shown as a function of matching frequency for different mass ratios. The $q=1$ line is the ratio between the 4 PN and 3.5 PN curves in Fig. 11.

Figure 14

Error analysis of PN-NR hybrids using only 3 PN information. The solid and dashed blue lines repeat the analysis of Fig. 5 at 3 PN order. The dotted red lines show $\Vert \delta h\Vert /\Vert h\Vert$ for hybrids using 3 PN waveforms compared to one using Taylor T4 at 3.5 PN (the circle is T1, the star T2, the square T3 and the downward triangle is T4). For reference, the data of Fig. 5 are included in grey in the background.