Frequency-domain gravitational waves from nonprecessing black-hole binaries. I. New numerical waveforms and anatomy of the signal

In this paper we discuss the anatomy of frequency-domain gravitational-wave signals from nonprecessing black-hole coalescences with the goal of constructing accurate phenomenological waveform models. We first present new numerical-relativity simulations for mass ratios up to 18, including spins. From a comparison of different post-Newtonian approximants with numerical-relativity data we select the uncalibrated SEOBNRv2 model as the most appropriate for the purpose of constructing hybrid post-Newtonian/numerical-relativity waveforms, and we discuss how we prepare time-domain and frequency-domain hybrid data sets. We then use our data together with results in the literature to calibrate simple explicit expressions for the final spin and radiated energy. Equipped with our prediction for the final state we then develop a simple and accurate merger-ringdown model based on modified Lorentzians in the gravitational-wave amplitude and phase, and we discuss a simple method to represent the low frequency signal augmenting the TaylorF2 post-Newtonian approximant with terms corresponding to higher orders in the post-Newtonian expansion. We finally discuss different options for modelling the small intermediate frequency regime between inspiral and merger ringdown. A complete phenomenological model based on the present work is presented in a companion paper [S. Khan et al., following paper, Phys. Rev. D 93 044007 (2016)].

Figure 1

Convergence and Richardson-extrapolated error for nonspinning BAM simulations at $q=18$ at resolutions (96,120,144) as described in the main text.

Figure 2

RMS difference between different resolutions as defined in Eq. (2.9) for nonspinning BAM simulations at $q=18$ at resolutions (96,120,144) as described in the main text. Rescaling for sixth-order convergence as shown in Fig. 1 is most consistent with our data.

Figure 3

Left: Orbital motion of the three highest resolutions (96, 120, 144) for the nonspinning $q=18$ case, aligned at $M{\omega }_{\mathrm{orb}}=0.2$. Right: Orbital motion of the two highest resolutions (96, 120) for the $q=18$, ${\chi }_{1,2}=0.4$, 0 case, aligned at $M{\omega }_{\mathrm{orb}}=0.15$. The point of alignment in tine is marked by a red circle, and an overall rotation is chosen for each numerical simulation to position the alignment point on the x axis.

Figure 4

Time shift $\mathrm{\Delta }t$ according to Eq. (2.16) as a function of wave frequency $M\omega$ for the case of nonspinning $q=1$, using SEOBNRv2 as the reference waveform and SXS:BBH:0001 from the SXS catalog as the NR waveform. As is well known, the SEOBNRv2 model and TaylorT4 are very close to the NR result in this case. SEOBv2 (uncalibrated SEOBNRv2) is also very close, while TaylorT1 (truncated at 3PN order) and TaylorT2 show significant dephasing. The SEOBNRv1 and SEOBv1 (uncalibrated SEOBNRv1) show rather similar performance, with good agreement with NR, and disagreement with SEOBNRv2 at lower frequencies.

Figure 5

Left panel: Time shift $\mathrm{\Delta }t$ according to Eq. (2.16) as a function of wave frequency $M\omega$ for the case of mass ratio 18 with spins $\{0.4,0\}$ with SEOBNRv2 as the reference waveform. The PN/EOB waveforms start at $M\omega =0.01$. Right panel: Same analysis for an equal mass binary with equal spins $\chi =0.98$. The NR waveform is SXS:BBH:0172 from the SXS catalog; see Table 3. SEOBNRv1/SEOBv1 are not shown for this case, since the model is not valid for high spins parallel to the orbital angular momentum.

Figure 6

Time domain amplitude of configurations at the corners of our parameter-space coverage, aligned at merger, showing both the NR waveform and the hybrid (marked as dashed lines). Thick black lines mark the hybridization regions.

Figure 7

Real part of the time-domain strain of the same corner cases as in Fig. 6, time coordinate is set to zero at the midfrequency of the hybridization region, which is shown in red.

Figure 8

Fourier-domain amplitude of same corner cases as in Fig. 6 and ringdown frequency. In the left panel, the thick lines indicate the part of the NR waveform that was used in the final hybrid, while data indicated by a thin line contained noise or other spurious artifacts and were discarded. The vertical lines indicate the peak of the amplitude and the ringdown frequency $f_{\mathrm{RD}}$. In the right panel the $f^{-7/6}$ leading-PN-order amplitude has been scaled out to illustrate detailed features of the Fourier-domain amplitude through merger and ringdown.

Figure 9

Real (oscillatory) and imaginary (damping) part of the ringdown frequency as a function of the final Kerr parameter $a_f$. A positive sign denotes a final BH parallel to the initial orbital angular momentum, and a negative sign denotes antiparallel.

Figure 10

Final Kerr parameter from Eq. (3.6) plotted as a function of symmetric mass ratio and total spin $\widehat{S}$. Black dots mark data points with equal spins and gray dots mark unequal spins.

Figure 11

Radiated energy according to fit Eq. (3.8) plotted as a function of symmetric mass ratio and total spin $\widehat{S}$. Black dots mark data points with equal spins and gray dots mark unequal spins.

Figure 12

Comparison of ringdown fit with numerical data for five “corner cases” as indicated in the legend. Full thin lines indicate raw numerical data, full thick lines indicate numerical data with a heuristic cutoff to remove noise at high frequencies, and dashed lines indicate the best fit to ansatz Eq. (4.5).

Figure 13

Residuals between numerical data and best fits for the data shown in Fig. 12.

Figure 14

Different forms of the TaylorF2 amplitude, rescaled as in the right panel of Fig. 8, are displayed for the $q=18$ case with the larger black hole spinning at ${\chi }_1=-0.8$. Shown are the numerical data of the hybrid waveform, the absolute value of the complex TaylorF2 amplitude from Eq. (4.1), and the real part of the TaylorF2 amplitude after reexpanding to $n=2.5$, 3, 3.5PN order. It can be seen that the reexpanded forms are closer to the hybrid data. Taking the real part or absolute value of expressions makes no significant difference for the case and frequency range shown.

Figure 15

Residuals for the inspiral fits corresponding to Eq. (4.8) with cutoff frequencies 0.016, 0.018, 0.02.

Figure 16

Fourier-domain phase corresponding to a hybrid constructed for SXS NR data corresponding to equal mass data with equal spins of ${\chi }_i=0.98$, with a term ${\varphi }_0+\omega t_0$ subtracted over a frequency range. The frequency ranges of the fits are indicated in the plot, where $M{f}_S=0.0035$ is the start frequency of clean hybrid data and $M{f}_{\mathrm{MR}}=0.018$ the frequency where we typically separate between inspiral and merger ringdown.

Figure 17

Phase derivative ${\varphi }^{\prime }$ for corner cases as indicated in the legend; vertical straight lines mark the ringdown frequency $f_{\mathrm{RD}}$.

Figure 18

Second phase derivative for the BAM waveform with $q=2$, ${\chi }_i=0.75$; vertical straight lines mark the ringdown frequency $f_{\mathrm{RD}}$ and $f_{\mathrm{RD}}\pm f_{\mathrm{damp}}/\sqrt{3}$.