New and robust gravitational-waveform model for high-mass-ratio binary neutron star systems with dynamical tidal effects

For the analysis of gravitational-wave signals, fast and accurate gravitational-waveform models are required. These enable us to obtain information on the system properties from compact binary mergers. In this article, we introduce the NRTidalv3 model, which contains a closed-form expression that describes tidal effects, focusing on the description of binary neutron star systems. The model improves upon previous versions by employing a larger set of numerical-relativity data for its calibration, by including high-mass ratio systems covering also a wider range of equations of state. It also takes into account dynamical tidal effects and the known post-Newtonian mass-ratio dependence of individual calibration parameters. We implemented the model in the publicly available lalsuite software library by augmenting different binary black hole waveform models (IMRPhenomD, IMRPhenomX, and SEOBNRv5_ROM). We test the validity of NRTidalv3 by comparing it with numerical-relativity waveforms, as well as other tidal models. Finally, we perform parameter estimation for GW170817 and GW190425 with the new tidal approximant and find overall consistent results with respect to previous studies.

Figure 1

Distribution of the masses (where we set $M_A>M_B$) and corresponding tidal deformabilities ${\mathrm{\Lambda }}_{A,B}$ of the neutron stars used in the NR simulations (see Table 7). The plot shows overlapping data points for systems with the same masses, but can have different tidal deformabilities due to the various EOSs that were employed.

Figure 2

An example of a hybrid waveform (SACRA:15H_135_135_00155_182_135). Here, we plot the real part of the (2,2) mode $r{h}_{22}$ of the GW strain (where $r$ is the extraction radius) as a function of the retarded time $u$. The blue curve is the NR waveform, the orange dashed curve is the EOB-BNS waveform from SEOBNRv4T, and the black, dotted curve is the hybridized waveform EOB-NR. The gray, dashed, vertical lines denote the boundaries of the hybridization window, $[t_1,t_2]$ corresponding to $\widehat{\omega }\in [0.035,0.04]$. The peak amplitude of the NR and EOB-NR hybrid, indicating merger, is set at $u/M=0$.

Figure 3

Hybrid tidal phases (EOB-NR) for the configuration SACRA:15H_135_135_00155_182_135. The cyan curve represents the EOB-NR data without applying the smoothening operation indicated in the text, the blue dashed line is EOB-NR after filtering. We also show the phase difference between the pure EOB and NR data, as well as for comparison the NRTidalv2 description. The reference frequency ${\widehat{\omega }}_{\mathrm{ref}}$ is used to subtract the EOB-BBH phase from the pure NR phase. The hybridization window $\widehat{\omega }\in [0.035,0.04]$ is denoted by the vertical dashed gray lines. Only the hybrid tidal phase up to the merger frequency ${\widehat{\omega }}_{\mathrm{mrg}}$ (indicated by a black vertical line) is used in the construction of NRTidalv3.

Figure 4

The universal relation between the $f$-mode frequency (in kHz) and $\mathrm{\Lambda }$ used for stars A and B.

Figure 5

Dynamical tidal parameter in the time-domain as a function of orbital frequency (${\widehat{\omega }}_{\mathrm{orb}}=\widehat{\omega }/2$) for different EOSs, given $q=1.0$. Note that each curve starts at the given constant value of the tidal parameter, which is provided by the dashed lines. The curves terminate at the merger frequency ${\widehat{\omega }}_{\mathrm{mrg}}=2{\widehat{\omega }}_{\mathrm{orb}}^{\mathrm{mrg}}$.

Figure 6

Time-domain tidal phase contributions calculated from different Models for the configuration of the NR waveform SACRA:15H_125_146_00155_182_135. We also show the EOB-NR phase for comparison. The vertical dashed line marks the merger frequency.

Figure 7

Top: The fits (dashed lines) plotted on top of the time-domain hybridized tidal phase data (solid lines). Bottom: Fractional differences between the fits and the data. The EOB-NR hybrid tidal phases are color-indicated by their EOS.

Figure 8

Top: The fits plotted on top of the frequency-domain hybridized tidal phase data. Bottom: Fractional differences between the fits and the data, where $\mathrm{\Delta }{\psi }_T={\psi }_T^{\text{Fit}}-{\psi }_T^{\text{Data}}$.

Figure 9

The difference between the absolute fractional differences of NRTidalv3 and NRTidalv2 with respect to the hybrid data. The curves for $q=1.0$ and $q>1.0$ are indicated. We observe NRTidalv3 to perform better (i.e., most curves are above the zero line) than NRTidalv2 for both equal and unequal mass ratios.

Figure 10

Frequency-domain, EOB-NR hybrid tidal phase contributions of SACRA:B_135_135_00155_182_135, SACRA:B_117_156_00155_182_135, BAM_0131, and BAM_0130, representing mass ratios [1.0, 1.33, 1.75, 2.0]. The tidal phase contributions for NRTidal, NRTidalv2, NRTidalv3, and 7.5 PN approximants are also shown. We also indicate the estimated merger frequency $f_{\mathrm{est}}$ from the fitting function in Sec. 4g, and the merger frequency of the NR simulations $f_{\mathrm{mrg}}$. The tidal contributions using other tidal models are also shown for comparison.

Figure 11

Time-domain dephasing comparisons for the BAM, SACRA, and SpEC waveforms. For each NR waveform, the upper panel shows the real part of the gravitational wave strain as a function of the retarded time, while the bottom panel shows the phase difference between the waveform model and the NR waveform. We note that BAM:0001, BAM:0037, and both SACRA waveforms (indicated by blue frames in their plots) were also used in the calibration of NRTidalv3.

Figure 12

Mismatch histograms for different BBH baselines augmented with the NRTidalv3 model, versus the BBH baseline model added with the EOB-NR hybrid tidal phase. Vertical lines denote the mean of the histograms with the same color. The mismatches with NRTidalv2 are also shown for comparison. For each BBH $\mathrm{baseline}+\mathtt{NRTidalv}\mathtt{3}$ model, mismatches are computed starting from $f_{\min }=20\mathrm{Hz}$ in the left panel, indicating the mismatch for the full waveform where EOB contribution is present in the early inspiral. Mismatches are also computed for $f_{\min }=350\mathrm{Hz}$ in the right panel, where typically the hybridization starts and where the NR contribution becomes dominant up to merger.

Figure 13

Mismatch Comparisons of NRTidalv3 with other tidal waveform models for nonspinning configurations. Each comparison with either TEOBResumS or SEOBNRv4T contains 4000 random configurations of mass $M_{A,B}=[0.5,3.0]$ and tidal deformability ${\mathrm{\Lambda }}_{A,B}=[0,5000]$. The rest of the comparisons are done with ${\mathrm{\Lambda }}_{A,B}=20000$. For each subfigure we include a color bar indicating the values of the mismatches. Note that the maximum value in the color bar for IMRPhenomXAS_NRTidalv3 vs SEOBNRv5_ROM_NRTidalv3 is different from the rest of the subfigures due to the NRTidalv3 models yielding very small mismatches with respect to each other.

Figure 14

Mismatch Comparisons of NRTidalv3 with other tidal waveform models for aligned-spin configurations. Each subfigure has 4000 random configurations of mass $M_{A,B}=[1.0,3.0]$, tidal deformability ${\mathrm{\Lambda }}_{A,B}=[0,5000]$ and spin ${\chi }_{A,B}=[-0.5,0.5]$. We also put the mismatches with NRTidalv2 for comparison. For each subfigure we include a color bar indicating the values of the mismatches.

Figure 15

Mismatches for IMRPhenomXP_NRTidalv3 vs IMRPhenomPv2_NRTidalv2 for precessing configurations. For this comparison, we generate 4000 random samples with mass $M_{A,B}=[0.5,3.0]$, tidal deformabilities ${\mathrm{\Lambda }}_{A,B}=[0,20000]$, and spins ${\chi }_{A,B,C}^{x,y,z}=[-0.5,0.5]$. For the spins, we plot the effective spin precession ${\chi }_p$ against the effective aligned spin ${\chi }_{\mathrm{eff}}$. We include a colorbar indicating the values of the mismatches.

Figure 16

The marginalized 1D and 2D posterior probability distributions for selected parameters of GW170817, obtained with IMRPhenomXP_NRTidalv3 (black), IMRPhenomPv2_NRTidalv2 (blue), and IMRPhenomXP_NRTidalv2 (orange). The parameters shown here are the individual star masses $M_{A,B}$, binary tidal deformability $\tilde{\mathrm{\Lambda }}$, luminosity distance $D$, and inclination angle ${\iota }_0$. The 68% and 90% confidence intervals are indicated by contours for the 2D posterior plots, while vertical lines in the 1D plots indicate 90% confidence interval. We note a narrow constraint on the tidal deformability for IMRPhenomXP_NRTidalv3 compared to the other models, due to the updated tidal information that was used.

Figure 17

Inferred spin parameters for GW170817 from a low-spin prior ($\chi \le 0.05$) using IMRPhenomXP_NRTidalv3. Plotted here are the probability densities for the dimensionless spin components ${\chi }_1$ (left hemisphere) and ${\chi }_2$ (right hemisphere) relative to the orbital angular momentum $\mathbf{L}$ and tilt angles (a tilt angle 0° means that the spin is aligned with the $\mathbf{L}$). The plot was done using a reference frequency of 20 Hz.

Figure 18

The marginalized 1D and 2D posterior probability distributions for selected parameters of GW190425, obtained with IMRPhenomXP_NRTidalv3 (black), IMRPhenomPv2_NRTidalv2 (blue), and IMRPhenomXP_NRTidalv2 (orange). The parameters shown here are the individual star masses $M_{A,B}$, binary tidal deformability $\tilde{\mathrm{\Lambda }}$, luminosity distance $D$, and inclination angle ${\iota }_0$. The 68% and 90% confidence intervals are indicated by contours for the 2D posterior plots, while vertical lines in the 1D plots indicate 90% confidence interval. As with GW170817, note a narrow constraint on the tidal deformability for IMRPhenomXP_NRTidalv3 compared to the other models, due to the updated tidal information that was used.

Figure 19

Inferred spin parameters for GW190425 from a low-spin prior ($\chi \le 0.05$) using IMRPhenomXP_NRTidalv3. Plotted here are the probability densities for the dimensionless spin components ${\chi }_1$ (left hemisphere) and ${\chi }_2$ (right hemisphere) relative to the orbital angular momentum $\mathbf{L}$ and tilt angles (i.e., a tilt angle 0° means that the spin is aligned with the $\mathbf{L}$). The plot was done at a reference frequency of 20 Hz.

Figure 20

The distribution of the log-likelihood $\ln \mathcal{L}$ for GW170817 (top) and GW190425 (bottom). The vertical lines indicate 90% confidence intervals for the models. We note the shifting of the distribution using IMRPhenomXP_NRTidalv3 towards larger $\ln \mathcal{L}$.

Figure 21

Top: The ${\overline{k}}_{2,\mathrm{rep}}^{\mathrm{eff}}$ plotted on top of the numerically calculated ${\overline{k}}_2^{\mathrm{eff}}$, color-coded according to EOS. Bottom: Fractional differences between the numerical calculation and the fit.

Figure 22

The marginalized 1D and 2D posterior probability distributions for selected parameters of GW170817, with a high-spin prior ($\chi \le 0.5$) obtained with IMRPhenomXP_NRTidalv3 (black), IMRPhenomPv2_NRTidalv2 (blue), and IMRPhenomXP_NRTidalv2 (orange). The parameters shown here are the individual star masses $M_{A,B}$, binary tidal deformability $\tilde{\mathrm{\Lambda }}$, luminosity distance $D$, and inclination angle ${\iota }_0$. The 68% and 90% confidence intervals are indicated by contours for the 2D posterior plots, while vertical lines in the 1D plots indicate 90% confidence interval. We note a narrow constraint on the tidal deformability for IMRPhenomXP_NRTidalv3 compared to the other models, due to the updated tidal information that was used.

Figure 23

Inferred spin parameters for GW170817 from a high-spin prior ($\chi \le 0.5$) using IMRPhenomXP_NRTidalv3. Plotted here are the probability densities for the dimensionless spin components ${\chi }_1$ (left hemisphere) and ${\chi }_2$ (right hemisphere) relative to the orbital angular momentum $\mathbf{L}$ and tilt angles (i.e., a tilt angle 0° means that the spin is aligned with the $\mathbf{L}$). The plot was done at a reference frequency of 20 Hz.

Figure 24

The marginalized 1D and 2D posterior probability distributions for selected parameters of GW190425, obtained with IMRPhenomXP_NRTidalv3 (black), IMRPhenomPv2_NRTidalv2 (blue), and IMRPhenomXP_NRTidalv2 (orange), for a high-spin prior ($\chi \le 0.50$). The parameters shown here are the individual star masses $M_{A,B}$, binary tidal deformability $\tilde{\mathrm{\Lambda }}$, luminosity distance $D$, and inclination angle ${\iota }_0$. The 68% and 90% confidence intervals are indicated by contours for the 2D posterior plots, while vertical lines in the 1D plots indicate 90% confidence interval. We note a narrow constraint on the tidal deformability for IMRPhenomXP_NRTidalv3 compared to the other models, due to the updated tidal information that was used.

Figure 25

Inferred spin parameters for GW190425 from a high-spin prior ($\chi \le 0.50$) using IMRPhenomXP_NRTidalv3. Plotted here are the probability densities for the dimensionless spin components ${\chi }_1$ (left hemisphere) and ${\chi }_2$ (right hemisphere) relative to the orbital angular momentum $\mathbf{L}$ and tilt angles (i.e., a tilt angle 0° means that the spin is aligned with the $\mathbf{L}$). The plot was done at a reference frequency of 20 Hz.

Figure 26

The distribution of the log-likelihood $\ln \mathcal{L}$ for GW170817 (top) and GW190425 (bottom) with high-spin priors ($\chi \le 0.5$). The vertical lines indicate 90% confidence intervals for the models. We note the shifting of the distribution using IMRPhenomXP_NRTidalv3 towards larger $\ln \mathcal{L}$.