Matching post-Newtonian and numerical relativity waveforms: Systematic errors and a new phenomenological model for nonprecessing black hole binaries

We present a new phenomenological gravitational waveform model for the inspiral and coalescence of nonprecessing spinning black hole binaries. Our approach is based on a frequency-domain matching of post-Newtonian inspiral waveforms with numerical relativity based binary black hole coalescence waveforms. We quantify the various possible sources of systematic errors that arise in matching post-Newtonian and numerical relativity waveforms, and we use a matching criteria based on minimizing these errors; we find that the dominant source of errors are those in the post-Newtonian waveforms near the merger. An analytical formula for the dominant mode of the gravitational radiation of nonprecessing black hole binaries is presented that captures the phenomenology of the hybrid waveforms. Its implementation in the current searches for gravitational waves should allow cross-checks of other inspiral-merger-ringdown waveform families and improve the reach of gravitational-wave searches.

Figure 1

These figures demonstrate the strain waveform obtained by the frequency-domain division method of calculating $h$ from ${\Psi }_4$. We consider the NR simulation from data set #8 of Table  with $q=1$, and we start with the dominant mode of ${\Psi }_4$ from this simulation. The upper panel shows $\tilde{A}=r|\tilde{h}(f)|$ (where $r$ is the extraction radius) obtained by the frequency-domain division, and the lower panel shows $h_{+}(t)$. The window function employed by our inverse Fourier transform algorithm is responsible for the partial loss of the first cycle of the waveform. Nevertheless, a clean $|h(t)|$ during the rest of the inspiral is observed.

Figure 2

Different variants of constructing the PN Fourier amplitude in the stationary phase approximation for the equal-mass case. The labels explain how $(\pi /\ddot{\varphi }{)}^{1/2}$ is treated in (3.10). The thick curve shows data obtained by a numerical simulation in full general relativity which begins at $Mf\approx 0.008$. The straight gray line illustrates the restricted PN amplitude $|{\tilde{h}}^{22}|D_L=\pi \sqrt{2\eta /3}(\pi f{)}^{-7/6}$.

Figure 3

A contour plot for the fitting error $\Delta {\varphi }_0$ in the $(f_L,\Delta f)$ plane. Here $\eta$ is kept fixed to the NR value, and we optimize over ${\varphi }_0$ and $t_0$.

Figure 4

Dependence of the fitting errors in $\eta$, ${\varphi }_0$, and $t_0$ on the frequency window $(f_L,\Delta f)$. Note that there is a clear choice of $(f_L,\Delta f)\approx (0.0093,0.014)$ that optimizes the fit between the NR waveform and the PN waveforms with different $\eta$. For a binary of total mass $10M_{\odot }$, for which the last stable orbit happens at 440 Hz, this corresponds to frequencies $(f_L,\Delta f){|}_{10M_{\odot }}\approx (189,284)\mathrm{Hz}$. This indicates that the optimal window for matching should start at the lowest reliable frequency available from the NR waveform and extend roughly up to the last stable orbit, which usually quantifies the point when the PN approximation starts to break down. Moreover, the plot on the left shows that the accuracy in $\eta$ decreases slowly with different choices of $(f_L,\Delta f)$, assuring that small changes in these values do not lead to large errors in the hybrid construction. At the best-fit point, the accuracy in $\eta$ by this fitting procedure is better than ${10}^{-3}$.

Figure 5

Best-fit value of $\eta$ as a function of the start frequency $f_L$ of the matching window for the waveform which corresponds nominally to a mass-ratio $1:2$, i.e. ${\eta }_{\mathrm{NR}}=2/9=0.222\dots$; this is shown by a horizontal dashed line. The vertical dashed line at $M{f}_L=0.009$ is the start frequency of the NR waveform. A rectangle highlights the region of minimal fitting errors from Fig. 4. We see that the best determined values of $\eta$ are clearly less than ${\eta }_{\mathrm{NR}}$.

Figure 6

Distinguishability of hybrid waveforms that have been constructed varying some of the hybrid ingredients at a time. When indicated, the black/grey color code denotes that Initial/Advanced LIGO design curves have been used for the distance calculation. The horizontal lines are the lines of constant SNR (in fact it is $1/{\mathrm{SNR}}^2$); if the distance measure goes above them, then the waveforms can be distinguished from each other. The left panel shows the effect of constructing hybrids from llama equal-mass waveforms at different resolutions. We CCATIE consider the difference between the high-medium resolution waveforms and the high-low waveform resolutions. The central panel shows the effect of using NR waveforms produced with either bah or llama codes. The solid lines indicate the normalized distance in the equal-mass case; dashed lines show the case of mass-ratio $1:2$. The highest available resolution was always used. The panel on the right displays Initial LIGO’s ability to distinguish hybrid waveforms constructed from different PN approximants. This plot shows that the hybrids are not sufficient for detection at the $ϵ=0.03$ level [Eq. (4.9)] only for a small range of masses. Early match is a reference for matching 3PN or 3.5PN F2 at early frequencies to the long equal-mass spec waveform.

Figure 7

Illustration of the method for constructing PN-NR hybrid waveforms in the frequency domain. The data corresponds to an equal-mass binary with aligned spins ${\chi }_1={\chi }_2=-0.25$. The left panel shows the amplitude and the right panel displays the phase of the dominant $\ell =2$, $m=2$ mode of the GW complex strain $\tilde{h}(f)$. The dotted lines correspond to the TaylorF2 PN approximant and the dot-dashed curve is the NR data. The hybrid waveform is depicted in solid black and the matching points for amplitude and phase are indicated with a dashed line.

Figure 8

Fitting procedure for the amplitude, applied to the equal-mass, nonspinning case. The ${\gamma }_1$ term of Eq. (5.11) is introduced to follow the behavior of the amplitude in the premerger regime; whereas the Lorentzian curve correctly describes the post-merger. The two pieces are glued together in a smooth manner using tanh-windows.

Figure 9

Map of the phenomenological parameters to the physical parameters of the binary.

Figure 10

Overlaps and fitting factors between the hybrid waveform constructed according to the procedure described in Sec. 4d and the proposed phenomenological fit, using the design sensitivity curve of Advanced LIGO. The labels indicate the values of $\eta$, $\chi$ for some configurations. In the upper panel we plot $\mathcal{O}(\lambda )=\mathcal{A}(\lambda ,\lambda )$, i.e. we compute the ambiguity function (4.10) without maximizing over the parameters of the model waveform; this is a lower bound on the effectualness. The bottom panel shows the maximized overlaps, i.e. $\mathcal{A}(\lambda ,{\lambda }^{\prime })$; the maximum bias of the optimized ${\lambda }^{\prime }$ parameters is $\Delta \eta =5\times {10}^{-3}$, $\Delta \chi =5\times {10}^{-2}$, $\Delta M=3M_{\odot }$.

Figure 11

Upper panel: maximized overlaps $\mathcal{A}(\lambda ,{\lambda }^{\prime })$ between the NR data sets #4–7ab and the predicted phenomenological waveforms from our model for Advanced LIGO. The labels indicate the values of $\eta$, $\chi$ for some configurations. Note that the short duration of the NR data prevents us from computing overlaps at lower masses. The maximization in this case has been done over $\eta$ and $\chi$ keeping $M$ fixed, and the maximum bias on the maximized parameters is $\Delta \eta =6\times {10}^{-3}$, $\Delta \chi =5\times {10}^{-2}$.