Frequency domain reduced order model of aligned-spin effective-one-body waveforms with generic mass ratios and spins

I provide a frequency domain reduced order model (ROM) for the aligned-spin effective-one-body model “SEOBNRv2” for data analysis with second- and third-generation ground-based gravitational wave (GW) detectors. SEOBNRv2 models the dominant mode of the GWs emitted by the coalescence of black hole binaries. The large physical parameter space (dimensionless spins $-1\le {\chi }_i\le 0.99$ and symmetric mass ratios $0.01\le \eta \le 0.25$) requires sophisticated reduced order modeling techniques, including patching in the parameter space and in frequency. I find that the time window over which the inspiral-plunge and the merger-ringdown waveform in SEOBNRv2 are connected has a discontinuous dependence on the parameters when the spin parameter $\chi =0.8$ or the symmetric mass ratio $\eta \sim 0.083$. This discontinuity increases resolution requirements for the ROM. The ROM can be used for compact binary systems with total masses of $2M_{\odot }$ or higher for the Advanced LIGO design sensitivity and a 10 Hz lower cutoff frequency. The ROM has a worst mismatch against SEOBNRv2 of $\sim 1\%$, but in general mismatches are better than $\sim 0.1\%$. The ROM is crucial for key data analysis applications for compact binaries, such as GW searches and parameter estimation carried out within the LIGO Scientific Collaboration.

Figure 1

The 38 SXS nonspinning (blue) and spinning (orange) NR simulations used to calibrate SEOBNRv2 as a function of symmetric mass ratio $\eta$ and the spin parameter $\chi$. In addition, the Teukolsky waveforms are shown in red.

Figure 2

An example tensor product grid on a base patch covering the entire domain of interest in model parameters ${\lambda }_1,{\lambda }_2$ and a refinement patch.

Figure 3

The Fourier domain amplitude for four corner cases in the parameter space (left panel). The FFT of SEOBNRv2 (solid lines) and the ROM (dashed lines) are shown at a total mass of $20M_{\odot }$ for a fixed distance of 500 Mpc viewed face on. The SEOBNRv2 ringdown for extreme antialigned configurations with equal spins ${\chi }_i=-1$ terminates earlier than the beginning of the ringdown for ${\chi }_i=0.99$. The ringdown frequency of SEOBNRv2 as a function of the symmetric mass ratio $\eta$ and equal spin ${\chi }_1={\chi }_2$ (right panel).

Figure 4

The location of input waveforms in $\eta$ and one of the aligned spins for the low-frequency ROM. Both spin components use the same grid.

Figure 5

The grid of input waveforms in the symmetric mass ratio $\eta$ and the aligned spin on the larger BH ${\chi }_1$ for the high-frequency ROMs. The points for the ROM on the low ${\chi }_1$ patch are shown in blue and those for the ROM on the high ${\chi }_1$ patch in orange. The left panel shows the full domain, while the middle and right panel show the transition between low and high ${\chi }_1$ grids near equal mass and near the extreme mass-ratio boundary $\eta =0.01$ of the domain, respectively.

Figure 6

Speedup of waveform generation of the overall ROM compared to SEOBNRv2 for nonspinning configurations. The speedup only depends very weakly on the value of spins.

Figure 7

The comb size parameter $\mathrm{\Delta }{t}_{\text{match}}^{22}/M$ used in SEOBNRv2 is discontinuous over the parameter space as indicated by the contours. Superimposed are configurations with mismatches $>0.1\%$ as shown in Fig. 8. They lie exactly on lines $\chi =0.8$ and $\eta =10/121\approx 0.083$ where the comb size has a discontinuity.

Figure 8

Faithfulness mismatches over the parameter space as a function of the spin of the larger BH ${\chi }_1$ and the symmetric mass ratio $\eta$.

Figure 9

Mismatches between the overall ROM and SEOBNRv2 for aLIGO design with a low-frequency cutoff of 10 Hz. To facilitate comparisons between panels, the color bar scales are fixed at a mismatch of 1%. The worst mismatches at high mass lie on lines where the SEOBNRv2 waveform does not depend smoothly on parameters (see Sec. 5c). Histograms of the mismatches are shown in Fig. 11 (left panel).

Figure 10

Mismatches between the overall ROM and SEOBNRv2 for early aLIGO with a low-frequency cutoff of 30 Hz. To facilitate comparisons between panels, the color bar scales are fixed at a mismatch of 1%. As in Fig. 9 the worst mismatches at high mass lie on lines where the SEOBNRv2 waveform does not depend smoothly on parameters (see Sec. 5c). Histograms of the mismatches are shown in Fig. 11 (right panel).

Figure 11

Mismatch histograms for aLIGO design 10 Hz (left) and aLIGO early 30 Hz (right). At high total mass isolated points have mismatch close to (or above) 1%. These configurations are due to nonsmooth parameter dependence in SEOBNRv2.

Figure 12

Mismatches between the overall ROM and SEOBNRv2 for ET-D with a low-frequency cutoff of 5 Hz.

Figure 13

Mismatches between the overall ROM and SEOBNRv2 for ET-D with a low-frequency cutoff of 2 Hz.

Figure 14

Offset between the time where the amplitude in the time domain peaks for SEOBNRv2 and ROM. If the offset is negative (positive), the peak in the ROM precedes (follows) the peak in SEOBNRv2. The offset is given in milliseconds for configurations at a total mass of $100M_{\odot }$ as a function of the symmetric mass ratio $\eta$ and equal aligned spins ${\chi }_i$.