Uncertainty in hybrid gravitational waveforms: Optimizing initial orbital frequencies for binary black-hole simulations

A general method is presented for estimating the uncertainty in hybrid models of gravitational waveforms from binary black-hole systems with arbitrary physical parameters, and thence the highest allowable initial orbital frequency for a numerical-relativity simulation such that the combined analytical and numerical waveform meets some minimum desired accuracy. The key strength of this estimate is that no prior numerical simulation in the relevant region of parameter space is needed, which means that these techniques can be used to direct future work. The method is demonstrated for a selection of extreme physical parameters. It is shown that optimal initial orbital frequencies depend roughly linearly on the mass of the binary, and therefore useful accuracy criteria must depend explicitly on the mass. The results indicate that accurate estimation of the parameters of stellar-mass black-hole binaries in Advanced LIGO data or calibration of waveforms for detection will require much longer numerical simulations than are currently available, or more accurate post-Newtonian approximations—or both—especially for comparable-mass systems with high spin.

Figure 1

Error caused by aligning at late times. This plot shows the phase error in hybrid waveforms created by aligning an ersatz NR waveform (EOB) to a PN model at various times, relative to the hybrid when aligned very far in the past. As the alignment point is moved closer to merger ($t/M=0$), the total phase error increases because the hybrid waveform incorporates more and more of the inaccurate PN data. Note that, in each case, the error grows most rapidly near merger. If the merger occurs at a frequency to which the detector is sensitive, this phase error will negatively impact the match found by data analysts (see Sec. 2a). Evaluating that impact using artificial NR data is the essence of the method presented in this paper.

Figure 2

Waveforms in the frequency domain. Amplitudes from an equal-mass nonspinning binary are shown, scaled to total system masses of $10M_{\odot }$ and $100M_{\odot }$ at $100\mathrm{Mpc}$, and compared to the noise spectral density of Advanced LIGO (both quantities in units of strain ${\mathrm{Hz}}^{-1/2}$). The factors multiplying the amplitudes are chosen to account for the logarithmic scaling of the horizontal axis, the factor of 2 in Eq. (1), and the fact that only positive frequencies are plotted. The triangles on the waveforms show the approximate initial frequency of the longest numerical simulation currently available. The circles show the frequency of the innermost stable circular orbit—basically the frequency at which PN approximations are expected to be useless. This plot shows that, for the $10M_{\odot }$ system, nearly half the contribution to the inner product of Eq. (1) comes from the PN data (to the left of the triangle), and the rest from the NR data (to the right). For the $100M_{\odot }$ system, on the other hand, the inner product is given almost exclusively by NR data.

Figure 3

Mismatches between hybrids as a function of ${\omega }_{\mathrm{hyb}}$. This plot shows the mismatch between pairs of hybrids using different approximants, for the equal-mass nonspinning system with $M_{\mathrm{tot}}=20M_{\odot }$. At any particular value of ${\omega }_{\mathrm{hyb}}$, the maximum mismatch between each pair of hybrids is the uncertainty in the final waveform hybridized at that frequency. If our target accuracy were, for example, $\mathrm{MM}\le {10}^{-3}$ for this system mass, this plot shows us that the NR waveform would need to contain the gravitational-wave frequency of $G{M}_{\mathrm{tot}}{\omega }_{\mathrm{hyb}}/c^3=0.02$, which naturally implies the initial orbital frequency of the simulation, ${\Omega }_0$. Achieving a smaller uncertainty requires lower ${\omega }_{\mathrm{hyb}}$, corresponding to times at which the PN approximants agree more closely.

Figure 4

Maximum mismatch between plausible hybrids for a selection of systems. These plots show the maximum mismatch between any pair of hybrids formed by hybridizing TaylorT1–T4 or EOB with an ersatz NR waveform (EOB) at ${\omega }_{\mathrm{hyb}}$, as described in Sec. 3a, when scaled to a total system mass of $M_{\mathrm{tot}}$. This quantity describes the uncertainty in the models used to form the hybrid. The plots show data from four cases with mass ratios denoted by $q$, which is the ratio of the larger to smaller mass, and components of the dimensionless spins $\chi$ aligned with the orbital angular-momentum vector. Note that $M_{\mathrm{tot}}\lesssim 33M_{\odot }$ may not be interesting astrophysically for black-hole binaries when $q=10$. The region $G{M}_{\mathrm{tot}}{\omega }_{\mathrm{hyb}}/c^3\gtrsim 0.055$ is inaccessible in the $q=10$, $\chi =0$ case, because the TaylorT3 approximant ends at that frequency; this could be interpreted as complete uncertainty. Generally, the uncertainty is larger in systems with more extreme parameters. The dotted red line in each plot shows $f_{\mathrm{seismic}}=10\mathrm{Hz}$, the lower bound of sensitivity in Advanced LIGO. The dashed green line in each plot shows $f_{20\%}$—the twentieth percentile of the power of the match, defined by Eq. (12). Comparing any plot to an accuracy requirement, which may depend on both $M_{\mathrm{tot}}$ and ${\omega }_{\mathrm{hyb}}$, we can extract the maximum sufficient hybridization frequency, which suggests the optimal initial orbital frequency. The method for doing this is described in Sec. 4. These plots are discussed in more detail in Sec. 4c.

Figure 5

Ratio of maximum mismatch to target mismatch. This plot shows the ratio between the maximum mismatch between various hybrids and a target mismatch given by Eq. (13). For values of this ratio greater than 1, the hybridization frequency is too high to achieve the target accuracy. The optimal sufficient value of ${\omega }_{\mathrm{hyb}}$ is given by the lowest frequency at which the ratio is 1—roughly $G{M}_{\mathrm{tot}}{\omega }_{\mathrm{hyb}}/c^3=0.011$ here.

Figure 6

Using real NR data for merger and ringdown. This plot reproduces the top-left plot of Fig. 4, using real NR data in place of the EOB ersatz NR waveform. The NR data start at a frequency $\omega \approx 0.035$, and are extended to lower frequencies by hybridizing with a TaylorT4 waveform. The plots are almost identical, indicating that the procedure is not strongly sensitive to details of the ersatz NR waveform.

Figure 7

Mismatch between the EOB waveform and the NR hybrid. This plot shows the mismatch as a function of total mass between the EOB waveform used in the body of this paper and the NR hybrid used in Fig. 6. The similarity between Figs. 4, 6, despite the significant mismatches shown here, leads us to conclude that the uncertainties shown in those figures are indeed robust with respect to the choice of ersatz NR waveform.