Exploring tidal effects of coalescing binary neutron stars in numerical relativity. II. Long-term simulations

We perform new long-term (15–16 orbits) simulations of coalescing binary neutron stars in numerical relativity using an updated Einstein equation solver, employing low-eccentricity initial data, and modeling the neutron stars by a piecewise polytropic equation of state. A convergence study shows that our new results converge more rapidly than the third order, and using the determined convergence order, we construct an extrapolated waveform for which the estimated total phase error should be less than one radian. We then compare the extrapolated waveforms with those calculated by the latest effective-one-body (EOB) formalism in which the so-called tidal deformability, higher post-Newtonian corrections, and gravitational self-force effects are taken into account. We show that for a binary of compact neutron stars with their radius 11.1 km, the waveform by the EOB formalism agrees quite well with the numerical waveform so that the total phase error is smaller than one radian for the total phase of $\sim 200$ radian up to the merger. By contrast, for a binary of less compact neutron stars with their radius 13.6 km, the EOB and numerical waveforms disagree with each other in the last few wave cycles, resulting in the total phase error of approximately three radian.

Figure 1

The waveform (real part; left) and amplitude (right) of ${\mathrm{\Psi }}_4^{2,2,\infty }(r_0,t_{\text{ret}})$ as functions of $t_{\text{ret}}$ for several values of $r_0$ for the run with H4 EOS and the best grid resolution ($N=72$). The lower plot of the left panel shows the phase differences of ${\mathrm{\Psi }}_4^{2,2,\infty }(r_0)$ relative to ${\mathrm{\Psi }}_4^{2,2,\infty }(r_0=237m_0)$.

Figure 2

The gravitational waveforms and the evolution of the gravitational wave phase for four different grid-resolution runs with the H4 EOS (left three panels) and with the APR4 EOS (right three panels). $N$ indicates the grid resolution, $\mathrm{\Delta }{x}_8\propto N^{-1}$. The upper panels show the gravitational waveforms for three different grid resolutions. The middle panels show the pure numerical wave phases and the bottom panels show the results obtained after the stretching of time and phase according to the convergence property (for $N=40$, 48, and 60). The lower plots in middle and bottom panels show the phase disagreement between the purely numerical wave phase for $N=72$ and the lower resolution results. Note that for $N=72$, $t_{\text{mrg}}=58.43(\mathrm{ms})$ for the H4 EOS and $t_{\text{mrg}}=61.08(\mathrm{ms})$ for the APR4 EOS.

Figure 3

The extrapolated gravitational waveform and related quantities for the models with the H4 (left) and APR4 EOS (right). (Top) The extrapolated waveforms for the best resolved ($N=72$) and second-best resolved ($N=60$) runs are plotted (two waveforms overlap quite well with each other and we cannot distinguish them in the figure). The waveform by an EOB calculation is plotted together. The lower panels focus on the late inspiral waveforms. (Middle) The extrapolated gravitational-wave frequency. In the lower panel of this, the absolute difference between the extrapolated result (with $N=72$) and EOB result is shown. (Bottom) The extrapolated gravitational-wave phase. In the lower panel of this, the difference between the extrapolated result and EOB result is shown. We aligned the phases of the extrapolated and EOB waveforms at $t_{\text{ret}}=5\mathrm{ms}$.

Figure 4

The evolution of the mismatch, $I_m(t_{\mathrm{r}\mathrm{e}\mathrm{t}})$, between the extrapolated waveform and EOB waveforms. “$\mathrm{EOB}\pm 3$” denotes that the EOB waveforms are employed artificially increasing or decreasing the neutron-star compactness by 3%.