Samurai project: Verifying the consistency of black-hole-binary waveforms for gravitational-wave detection

We quantify the consistency of numerical-relativity black-hole-binary waveforms for use in gravitational-wave (GW) searches with current and planned ground-based detectors. We compare previously published results for the $(\ell =2,|m|=2)$ mode of the gravitational waves from an equal-mass nonspinning binary, calculated by five numerical codes. We focus on the $1000M$ (about six orbits, or 12 GW cycles) before the peak of the GW amplitude and the subsequent ringdown. We find that the phase and amplitude agree within each code’s uncertainty estimates. The mismatch between the $(\ell =2,|m|=2)$ modes is better than ${10}^{-3}$ for binary masses above $60M_{\odot }$ with respect to the Enhanced LIGO detector noise curve, and for masses above $180M_{\odot }$ with respect to Advanced LIGO, Virgo, and Advanced Virgo. Between the waveforms with the best agreement, the mismatch is below $2\times {10}^{-4}$. We find that the waveforms would be indistinguishable in all ground-based detectors (and for the masses we consider) if detected with a signal-to-noise ratio of less than $\approx 14$, or less than $\approx 25$ in the best cases.

Figure 1

Theoretical noise curves [power spectral density $S_n$; see Eq. (12)] for the Enhanced LIGO, Advanced LIGO, Virgo, and Advanced Virgo detectors.

Figure 2

The gravitational-wave strain from an optimally oriented $60M_{\odot }$ equal-mass nonspinning black-hole binary located 100 Mpc away from the detector. The waveform covers about six orbits, or 12 GW cycles, before merger.

Figure 3

The GW frequency as calculated from the raw $\mathtt{B}\mathtt{A}\mathtt{M}$ data, and as given by the fitting procedure described in the text. The two lines are indistinguishable if viewed in black-and-white.

Figure 4

GW phase $\varphi$ as a function of frequency $M\omega$, for the five codes. The frequency is given both in terms of the dimensionless orbital frequency $M\omega$ and the frequency in kHz scaled with respect to the total mass of the binary in solar masses.

Figure 5

Phase comparison. The left panel shows the phase comparison between the $\mathtt{S}\mathtt{p}\mathtt{E}\mathtt{C}$ waveform and the others during inspiral, from $M\omega =0.055$ up to $M\omega =0.2$, which is about one orbit before merger. The corresponding uncertainty estimates are 0.1 rad ($\mathtt{B}\mathtt{A}\mathtt{M}$ and $\mathtt{M}\mathtt{a}\mathtt{y}\mathtt{a}\mathtt{K}\mathtt{r}\mathtt{a}\mathtt{n}\mathtt{c}$) and 2.4 rad ($\mathtt{H}\mathtt{a}\mathtt{h}\mathtt{n}\mathtt{d}\mathtt{o}\mathtt{l}$). The right panel shows the phase comparison during merger and ringdown, from $M\omega =0.2$ up to $M\omega =0.55$. The uncertainty estimates during merger are, in radians, 1.0 ($\mathtt{B}\mathtt{A}\mathtt{M}$), 1.1 ($\mathtt{M}\mathtt{a}\mathtt{y}\mathtt{a}\mathtt{K}\mathtt{r}\mathtt{a}\mathtt{n}\mathtt{c}$), and 5.0 ($\mathtt{H}\mathtt{a}\mathtt{h}\mathtt{n}\mathtt{d}\mathtt{o}\mathtt{l}$). In both panels a phase shift was applied so that the phases agreed at the lowest frequency shown.

Figure 6

The amplitude as a function of GW frequency, $A(\omega )=|r{\Psi }_{4,22}|$, for the five codes.

Figure 7

Comparison of the amplitude as a function of GW frequency, $A(\omega )$. The left panel shows the percentage disagreement during inspiral (up to $M\omega =0.2$). The corresponding amplitude uncertainties are 2% ($\mathtt{C}\mathtt{C}\mathtt{A}\mathtt{T}\mathtt{I}\mathtt{E}$), 4% ($\mathtt{B}\mathtt{A}\mathtt{M}$ and $\mathtt{M}\mathtt{a}\mathtt{y}\mathtt{a}\mathtt{K}\mathtt{r}\mathtt{a}\mathtt{n}\mathtt{c}$), and 10% ($\mathtt{H}\mathtt{a}\mathtt{h}\mathtt{n}\mathtt{d}\mathtt{o}\mathtt{l}$). The right panel shows the same quantity during merger and ringdown, for which the uncertainties are 5% ($\mathtt{C}\mathtt{C}\mathtt{A}\mathtt{T}\mathtt{I}\mathtt{E}$), 6% ($\mathtt{B}\mathtt{A}\mathtt{M}$), 8% ($\mathtt{M}\mathtt{a}\mathtt{y}\mathtt{a}\mathtt{K}\mathtt{r}\mathtt{a}\mathtt{n}\mathtt{c}$), and 10% ($\mathtt{H}\mathtt{a}\mathtt{h}\mathtt{n}\mathtt{d}\mathtt{o}\mathtt{l}$). The vertical dashed line indicates the approximate location of the amplitude maximum, $M\omega =0.5$.

Figure 8

The mismatch between the $\mathtt{S}\mathtt{p}\mathtt{E}\mathtt{C}$ waveform and each of the other codes (the line colors match those in previous plots). The three plots show the results for the Enhanced LIGO, Advanced LIGO, Virgo, and Advanced Virgo noise curves. The lower end of the mass range was chosen such that the entire numerical waveform was included in the detector’s frequency band.

Figure 9

The SNR below which the $\mathtt{S}\mathtt{p}\mathtt{E}\mathtt{C}$ and all other waveforms will be indistinguishable in any measurement of parameters. Results are shown for the Enhanced LIGO, Advanced LIGO, Virgo, and Advanced Virgo detectors. See text for further explanation.