Gravitational waveforms from spectral Einstein code simulations: Neutron star-neutron star and low-mass black hole-neutron star binaries

Gravitational waveforms from numerical simulations are a critical tool to test and analytically calibrate the waveform models used to study the properties of merging compact objects. In this paper, we present a series of high-accuracy waveforms produced with the spectral Einstein code (SpEC) for systems involving at least one neutron star. We provide for the first time waveforms with subradian accuracy over more than twenty cycles for low-mass black hole-neutron star binaries, including binaries with nonspinning objects, and binaries with rapidly spinning neutron stars that maximize the impact on the gravitational wave signal of the near-resonant growth of the fundamental excitation mode of the neutron star (f-mode). We also provide for the first time with SpEC a high-accuracy neutron star-neutron star waveform. These waveforms are made publicly available as part of the SxS catalogue. We compare our results to analytical waveform models currently implemented in data analysis pipelines. For most simulations, the models lie outside of the predicted numerical errors in the last few orbits before merger, but do not show systematic deviations from the numerical results: comparing different models appears to provide reasonable estimates of the modeling errors. The sole exception is the equal-mass simulation using a rapidly counterrotating neutron star to maximize the impact of the excitation of the f-mode, for which all models perform poorly. This is however expected, as even the single model that takes f-mode excitation into account ignores the significant impact of the neutron star spin on the f-mode excitation frequency.

Figure 1

Numerical error in the phase of the (2,2) mode of the GW signal for the 6 simulations using a $\mathrm{\Gamma }$-law equation of state. For each simulation, we show estimates of the discretization error (dashed blue), mass loss error (dashed red) and extrapolation error (dashed green), as well as the total numerical error (solid black line) defined by Eq. (1). The vertical dashed line shows the time of maximum amplitude of the waveform.

Figure 2

Same as Fig 1, but after allowing for an arbitrary time and phase shift in the low-resolution results of case BHNSq1.5s0, minimizing phase errors in the time interval [1000, 1700].

Figure 3

Same as Fig. 1, but the BHNS binary using the H1 equation of state.

Figure 4

Same as Fig. 1, but for the NSNS simulation using the MS1b equation of state. In this case, the numerical error is nearly entirely due to the effect of unresolved transients at early times.

Figure 5

Dominant (2,2) mode of the gravitational wave signal for all $q=1$ cases using the $\mathrm{\Gamma }2$ equation of state. The shaded regions in the zoom-in around merger time (bottom panel) lie in between waveforms dephased by the estimated errors from Fig. 7. The waveform for the binary black hole simulation is assumed to be exact, as errors are significantly smaller for vacuum simulations than for simulations involving neutron stars. All waveforms are aligned through a time and phase shift minimizing the phase difference in the time interval $100<t/M<1100$.

Figure 6

Same as Fig. 5, but for the $q=2$ configurations. The errors in the bottom panel are from Fig. 8.

Figure 7

Phase difference between the (2,2) modes of the gravitational wave signals of the $q=1$ systems with $\mathrm{\Gamma }$-law equation of state, and an equal mass, nonspinning binary black hole waveform. The waveforms are aligned by applying a time and phase shifts minimizing the phase error in the time interval $100M<t<1100M$ of the nonspinning BHNS system. Dashed curves show our conservative estimate of the phasing error, aligned over the same time interval, and the vertical lines correspond to the time of peak gravitational wave amplitude for each system. We see that both tidal effects and spin effects are resolved in the simulations, conservatively within a few percents at the peak of the gravitational wave signal ($\sim 10\%$ if using raw numerical error without alignment).

Figure 8

Same as Fig. 7, but for the asymmetric $q=2$ BHNS systems, both compared to a nonspinning $q=2$ binary black hole system. As finite-size effects are smaller, and errors larger, we can only guarantee that tidal and spin effects are resolved at the $\sim 25\%$ level at the peak of the gravitational wave signal (with or without alignment of the waveforms).

Figure 9

Comparison between numerical waveforms and analytical models for nonspinning BHNS binaries. For each configuration, the left panel shows the amplitude of the “$+$” polarization of the dominant (2,2) mode of the gravitational wave signal, zooming in on the region where models and simulations diverge (the gray curves are numerical results, while other curves are model predictions). The right panel shows phase differences between analytical models and the highest resolution numerical waveform at our disposal. In that panel, solid lines denote regions where the analytical model is ahead of the simulation, and dashed lines regions where the simulation is ahead of the model. The dashed vertical line in the right panel corresponds to the peak of the GW signal. LEA is the only model used here that attempts to capture the waveform past that peak.

Figure 10

Same as Fig. 9, but for BHNS systems with spinning neutron stars, and for the equal mass NSNS system with $\mathrm{\Gamma }2$ equation of state. As before, dashed and solid curves denote phase errors of different signs.