Approaching the Post-Newtonian Regime with Numerical Relativity: A Compact-Object Binary Simulation Spanning 350 Gravitational-Wave Cycles

We present the first numerical-relativity simulation of a compact-object binary whose gravitational waveform is long enough to cover the entire frequency band of advanced gravitational-wave detectors, such as LIGO, Virgo, and KAGRA, for mass ratio 7 and total mass as low as $45.5M_{\odot }$. We find that effective-one-body models, either uncalibrated or calibrated against substantially shorter numerical-relativity waveforms at smaller mass ratios, reproduce our new waveform remarkably well, with a negligible loss in detection rate due to modeling error. In contrast, post-Newtonian inspiral waveforms and existing calibrated phenomenological inspiral-merger-ringdown waveforms display greater disagreement with our new simulation. The disagreement varies substantially depending on the specific post-Newtonian approximant used.

Figure 1

Overview of the new very long simulation. The main panel shows the (2,2) spherical-harmonic mode of the GW strain, with enlargements in the lower insets. The top inset shows the Fourier spectra of the new waveform in blue and the NR-NR hybrid waveform (used for comparisons with analytical models) in yellow, overlaid with noise power-spectral densities of aLIGO at the early (dashed black) and design (solid black) sensitivity [4]. The waveforms in the inset are scaled to total mass $M=45.5M_{\odot }$ and luminosity distance $D_L\approx 1.06\mathrm{Gpc}$. For comparison, an older $q=6$ waveform [27] of representative length is shown in the main panel (offset vertically for clarity) and in the power-spectrum inset.

Figure 2

Top left: displacement of the c.m. from the origin for three long simulations with different outer boundary radii $R$. In each case, $|{\overrightarrow{c}}_{\mathrm{c}.\mathrm{m}.}|$ increases exponentially. Top right: growth rate $\sigma$ of $|{\overrightarrow{c}}_{\mathrm{c}.\mathrm{m}.}|$ as a function of $R$ with a power-law fit. Lower panels: GW (2,2) mode of the short and long simulations. The long simulation still agrees very well with the short one at early times but fails to produce an accurate merger waveform.

Figure 3

Unfaithfulness of the hybrid NR-NR waveform against analytical models. Left: inspiral-only comparisons. Right: IMR comparisons. Also shown in the left panel are ${\mathrm{SNR}}_{\text{insp}}^2/{\mathrm{SNR}}_{\text{full–insp}}^2$ (solid black line) and ${\mathrm{SNR}}_{\text{insp}}^2/{\mathrm{SNR}}_{\text{full}–\mathrm{IMR}}^2$ (dot-dashed black line), and in the right panel ${\mathrm{SNR}}_{\mathrm{IMR}}^2/{\mathrm{SNR}}_{\text{full}–\mathrm{IMR}}^2$ (solid black line). The blue area indicates the NR error. We note that the “EOB curve” does not use any information from NR simulations and the “EOBNR curves” were calibrated to other, much shorter NR waveforms with $q\ne 7$, but not to the NR waveforms of this paper.