Accurate numerical simulations of inspiralling binary neutron stars and their comparison with effective-one-body analytical models

Binary neutron-star systems represent one of the most promising sources of gravitational waves. In order to be able to extract important information, notably about the equation of state of matter at nuclear density, it is necessary to have in hand an accurate analytical model of the expected waveforms. Following our recent work [L. Baiotti, T. Damour, B. Giacomazzo, A. Nagar, and L. Rezzolla, Phys. Rev. Lett. 105, 261101 (2010).], we here analyze more in detail two general-relativistic simulations spanning about 20 gravitational-wave cycles of the inspiral of equal-mass binary neutron stars with different compactnesses, and compare them with a tidal extension of the effective-one-body (EOB) analytical model. The latter tidally extended EOB model is analytically complete up to the 1.5 post-Newtonian level, and contains an analytically undetermined parameter representing a higher-order amplification of tidal effects. We find that, by calibrating this single parameter, the EOB model can reproduce, within the numerical error, the two numerical waveforms essentially up to the merger. By contrast, analytical models (either EOB or Taylor-T4) that do not incorporate such a higher-order amplification of tidal effects, build a dephasing with respect to the numerical waveforms of several radians.

Figure 1

Curvature gravitational waveforms $r{\psi }_4^{22}$ (upper panels) and their instantaneous frequency $M{\omega }_{22}$ (lower panels) for the $\mathtt{M}\mathtt{2.9}\mathtt{C}\mathtt{.}\mathtt{12}$ (left) and $\mathtt{M}\mathtt{3.2}\mathtt{C}\mathtt{.}\mathtt{14}$ (right) models. In both cases, the observer’s (coordinate) extraction radius is $r_{\mathrm{obs}}=500M_{\odot }$; this corresponds to $r_{\mathrm{obs}}/M=184.3$ for $\mathtt{M}\mathtt{2.9}\mathtt{C}\mathtt{.}\mathtt{12}$ and $r_{\mathrm{obs}}/M=165.1$ for $\mathtt{M}\mathtt{3.2}\mathtt{C}\mathtt{.}\mathtt{14}$.

Figure 2

Metric gravitational waveforms $r{h}_{22}$ and frequencies (upper panels) and the corresponding instantaneous frequency $M{\omega }_{22}$ (lower panels) obtained from the (double) time-integration of the curvature waveforms of Fig. 1 [see Eq. (6)]. The left panels refer to model $\mathtt{M}\mathtt{2.9}\mathtt{C}\mathtt{.}\mathtt{12}$, the right panels to model $\mathtt{M}\mathtt{3.2}\mathtt{C}\mathtt{.}\mathtt{14}$. The fact that the waveform modulus grows monotonically without evident spurious oscillations is the indication of the reliability of the determination of the integration constants. See text for details.

Figure 3

Exploring the properties of $Q_{\omega }$ curves computed within the EOB model for three binary systems. Tidal interactions are approximated at LO. The inset shows a magnification, in order to highlight the differences among the curves.

Figure 4

Obtaining the $Q_{\omega }$ diagnostic from a suitable fitting procedure of the GW phase (for both curvature and metric waveforms). The two vertical lines on the left panels indicate the time-interval $\Delta t/M=[1000,2290]$ where we fit the NR phase with Eq. (28). For completeness we also display the real part of the metric waveform. On the right panels, the (light) dashed lines refer to the $Q_{\omega }$ obtained by direct numerical differentiation of the raw data; the solid lines are instead obtained from the fitted phase. Although the curves displayed here refer to model $\mathtt{M}\mathtt{3.2}\mathtt{C}\mathtt{.}\mathtt{14}$, similar results are obtained also for the binary $\mathtt{M}\mathtt{2.9}\mathtt{C}\mathtt{.}\mathtt{12}$.

Figure 5

Comparing waveforms from isentropic (dashed) and nonisentropic (solid) evolution for BNS model $\mathtt{M}\mathtt{3.2}\mathtt{C}\mathtt{.}\mathtt{14}$. Waveforms are computed with the highest resolution and extracted at $r_{\mathrm{obs}}=500M_{\odot }$. The corresponding phase difference ${\varphi }^{{\mathrm{poly}}_{\mathrm{HR}}500}-{\varphi }^{{\mathrm{IF}}_{\mathrm{HR}}500}$ is displayed in Fig. 6.

Figure 6

Estimate of the phase uncertainty in the time domain for model $\mathtt{M}\mathtt{3.2}\mathtt{C}\mathtt{.}\mathtt{14}$ (top) and $\mathtt{M}\mathtt{2.9}\mathtt{C}\mathtt{.}\mathtt{12}$ (bottom). The figure shows the phase difference between different “post-processed” numerical curvature waveforms $r{\psi }_4^{22}$ (in particular, extrapolated in resolution and/or extraction radius) and the one obtained with the ideal-fluid EOS and extracted at $r_{\mathrm{obs}}=500M_{\odot }$.

Figure 7

Left panel: span of $Q_{\omega }$’s due to the various approximations to the curvature waveforms from model $\mathtt{M}\mathtt{3.2}\mathtt{C}\mathtt{.}\mathtt{14}$. Right panel: the corresponding differences $\Delta Q_{\omega }=Q_{\omega }^X-Q_{\omega }^{{\mathrm{IF}}_{\mathrm{HR}}500}$ between the various curves and the fiducial one obtained from the phase computed at the highest resolution and extracted at $500M_{\odot }$.

Figure 8

Sensitivity of $Q_{\omega }$ to the phase model used in the fitting procedure. Note that the $n=4$ and $n=6$ curves are barely distinguishable on the plot. See text for details.

Figure 9

Sensitivity of $Q_{\omega }$ to the choice of the fitting time-interval $I_t$ for $\mathtt{M}\mathtt{3.2}\mathtt{C}\mathtt{.}\mathtt{14}$. Our preferred cleaning time-interval $I_t/M=[1000,2290]$ (central dashed line) is compared to $I_t/M=[1000,2250]$ (solid line) and $I_t/M=[1000,2330]$ (dash-dotted line). See text for details.

Figure 10

Subtraction of tidal effects: shown as a solid line is the point-mass EOB curve, while shown as a dashed line is the $Q_{\omega }^0$ curve obtained by inserting in Eq. (36) the tidally-modified EOB $Q_{\omega }$ curves shown in Fig. 3.

Figure 11

Subtraction of tidal effects from numerical relativity (curvature) $Q_{\omega }$ curves according to Eq. (36). Note the excellent agreement with the point-mass EOB curve in the frequency window where $\mathtt{M}\mathtt{2.9}\mathtt{C}\mathtt{.}\mathtt{12}$ and $\mathtt{M}\mathtt{3.2}\mathtt{C}\mathtt{.}\mathtt{14}$ data overlap. The relative EOB-NR phase difference accumulated over this overlap interval is $\Delta {\varphi }_{{\psi }_4}^{\mathrm{EOBNR}}=-0.03\mathrm{rad}$.

Figure 12

Comparison of the EOB $Q_{\omega }$ curves for different choices of the effective tidal amplification factor ${\widehat{A}}_{\ell }^{\mathrm{tidal}}(u)=1+{\overline{\alpha }}_{1}u+{\overline{\alpha }}_2{u}^2$, with the corresponding NR ones (dashed lines with open circles) for the two binaries considered. The dotted line corresponds to the “tidal-free” (or “point-mass”) EOB, namely, when ignoring tidal effects. The figure also includes the tidal-free Taylor-T4 model. The good visual agreement between the analytic and the numerical curves for ${\overline{\alpha }}_2=100$ provides evidence of the need for large NNLO tidal corrections. The corresponding phase differences $\Delta {\varphi }_{{\psi }_4}={\varphi }^{\mathrm{EOB}}-{\varphi }^{\mathrm{NR}}$ are listed in Table .

Figure 13

Magnitude of NNLO tidal effects: span of EOB $Q_{\omega }$ curves (red) with varying ${\overline{\alpha }}_2$ so as to be compatible with the various (numerical) $Q_{\omega }$ curves (black).

Figure 14

Comparison of the Taylor-T4 $Q_{\omega }$ curves for different choices of the effective tidal amplification factor ${\widehat{a}}^{\mathrm{tidal}}(u)=1+a_1^{\mathrm{T}4}x+a_2^{\mathrm{T}4}{x}^2$, with the corresponding NR ones (dashed lines with open circles) for the two binaries considered. The dotted line corresponds to the “tidal-free” (or “point-mass”) T4, namely, when ignoring tidal effects. Note that the value $a_2^{\mathrm{T}4}=350$ of the dimensionless NNLO effective tidal correction parameter that best matches the ($\mathtt{M}\mathtt{3.2}\mathtt{C}\mathtt{.}\mathtt{14}$) NR data is considerably larger than in the EOB case. The corresponding phase differences $\Delta {\varphi }_{{\psi }_4}={\varphi }^{\mathrm{T}4}-{\varphi }^{\mathrm{NR}}$ are listed in Table .

Figure 15

Comparison between EOB and NR phasing for the $\mathtt{M}\mathtt{2.9}\mathtt{C}\mathtt{.}\mathtt{12}$ (left panels) and $\mathtt{M}\mathtt{3.2}\mathtt{C}\mathtt{.}\mathtt{14}$ (right panels) binaries. The top panels show the real parts of the EOB and NR $h_{22}$ waveforms, the middle panels display the corresponding phase differences $\Delta {\varphi }^{\mathrm{EOBNR}}={\varphi }^{\mathrm{EOB}}-{\varphi }^{\mathrm{NR}}$, both metric (solid line) and curvature (dashed line) waveforms, the bottom panels compare the EOB (dashed line) and NR (solid line) instantaneous GW frequency. The NNLO corrections to the radial potential are carried out with the parameter ${\overline{\alpha }}_2=100$. Note the agreement reached with the numerical waveform almost up to the time of the merger as defined in terms of the maximum of the GW amplitude (vertical dashed line) or of the contact position (dot-dashed line; see the text for details).

Figure 16

Testing the fit of the GW phase of the $\mathtt{M}\mathtt{2.9}\mathtt{C}\mathtt{.}\mathtt{12}$ simulation. The top-left panel shows the time evolution of the frequency, computed from the metric and curvature waveforms. The bottom-left panel shows the deviation of the cleaned phase evolution with respect to the raw data; note that they average to zero. The right panels show the comparison of the frequency evolution of the cleaned and raw waveforms, for the curvature (top) and metric (bottom) waveforms.

Figure 17

The same as Fig. 16 but for the $\mathtt{M}\mathtt{3.2}\mathtt{C}\mathtt{.}\mathtt{14}$ simulation.