Effect of dynamical gravitomagnetic tides on measurability of tidal parameters for binary neutron stars using gravitational waves

Gravitational waves (GWs) from binary neutron stars (NSs) have opened unique opportunities to constrain the nuclear equation of state by measuring tidal effects associated with the excitation of characteristic modes of the NSs. This includes gravitomagnetic modes induced by the Coriolis effect whose frequencies are proportional to the NS’s spin frequency. The NS’s spin orientation determines which subclass of gravitomagnetic modes are predominantly excited. We incorporate these effects in GW models needed for data analysis by encapsulating the adiabatic signatures from gravitomagnetic modes in slowly rotating NSs in an effective Love number which differs before and after a mode resonance and combining this with a known generic model for abrupt changes in the GWs at the mode resonance. This leads to an efficient approximate model that adds to a point-mass baseline and which we use to perform case studies of the impacts of gravitomagnetic effects for measurements with Cosmic Explorer, an envisioned next-generation GW detector. We quantify the extent to which neglecting (including) the effect of gravitomagnetic modes induces biases (significantly reduces statistical errors) in the measured tidal deformability parameters, which depend on the equation of state. Our results substantiate the importance of dynamical gravitomagnetic tidal effects for measurements with third-generation detectors.

Figure 1

Phase accumulation due to tidal effects described in (3.1b) and (3.1c). The plot shows phase accumulation due to adiabatic electric tidal effects (blue curve), adiabatic magnetic tidal effects (orange and red curve), and resonant magnetic tidal effects (green and violet curve) as a function of frequency for aligned spins $\chi =\{0.005\mathrm{and}0.01\}$ with masses $(M_1,M_2)=(1.5,1.3)M_{\odot }$ and tidal deformability parameters $(\tilde{\mathrm{\Lambda }},\delta \tilde{\mathrm{\Lambda }})=(519.38,48.37)$. The terms “Electric,” “Mag. adi” and “Mag. res” denotes adiabatic gravitoelectric tidal contribution in (3.1b), adiabatic gravitomagnetic tidal contribution in (3.1b) and resonant gravitomagnetic tidal contribution in (3.1c) respectively.

Figure 2

CE noise spectral density and various mode resonances for the two bodies and varying spins $\chi$. Green and purple symbols refer to the NS with mass $M_1=1.5M_{\odot }$ with lower and higher spin respectively, while red and brown symbols are the corresponding values for the companion of mass $M_2=1.3M_{\odot }$. Diamond shapes denote the modes with azimuthal number $m=1$, triangles those with $m=2$. Note that the y value of the symbols has no meaning.

Figure 3

Posterior probability distribution of $\tilde{\mathrm{\Lambda }}$ for SNR 1800 with the PNTidal waveform model (without gravitomagnetic effects) used for injection and recovery. The label 4D refers to a reduced parameter space of the tidal deformabilities $\tilde{\mathrm{\Lambda }},\delta \tilde{\mathrm{\Lambda }}$ and the time and phase of coalescence $t_c,{\varphi }_c$ with all other parameters fixed, while 8D also includes the sampling of the mass and spin parameters for each body. The results from the Fisher matrix (orange curve) agree well with the corresponding Bayesian analysis (blue curve), with both centered on the injected value (vertical line). The green curve shows the broadening of the distribution when doubling the dimensionality of the parameter space sampled. The shaded tails of the curves indicate regions outside the 90% credible interval. For the parameter $\delta \tilde{\mathrm{\Lambda }}$, both the 4D and 8D posteriors are essentially flat in this case.

Figure 4

Shifts in the posterior distribution for $\tilde{\mathrm{\Lambda }}$ due to adiabatic and resonant gravitomagnetic effects. This case study is for SNR 1800, aligned spins $\chi =0.005$, and sampling only on $(\tilde{\mathrm{\Lambda }},t_c,{\varphi }_c)$ with all other parameters fixed. We inject with a waveform that includes all effects ${\mathrm{PNTidal}}_{\mathrm{asym}}^{\text{modes}}$ and recover with the same waveform (green curve) and those that include only the resonance jumps (blue curve) and only the adiabatic effects (orange curve). In this case the contribution from the adiabatic effects is dominant; omitting them (as for the results shown by the blue curve) leads to the largest shifts in the distribution.

Figure 5

Posterior distributions of the tidal parameters for nonspinning NSs at SNR 1800. The injection neglected gravitomagnetic tides, and the blue curve illustrates the recovery with the same waveform. The effect of gravitomagnetic tides, which are purely adiabatic in this case, is indicated by the orange curve. In the two-dimensional representation in the lower left panel, the contours correspond to the 1- and $2\text{−}\sigma$ confidence levels.

Figure 6

Gravitomagnetic effects for aligned spins of ${\chi }_{1,2}=0.005$ and SNR 1800. The blue curve corresponds to using the same waveform for injection and recovery. Comparing this with the orange curve indicates the changes due to gravitomagnetic tides from both the $m=1$ mode resonances and the adiabatic effects, which lead to a shift in the distribution of $\tilde{\mathrm{\Lambda }}$ and a slight change in the shape of the $\delta \tilde{\mathrm{\Lambda }}$ posterior.

Figure 7

Gravitomagnetic effects for aligned spins of ${\chi }_{1,2}=0.01$ and SNR 1800. The blue curve corresponds to using the same waveform for injection and recovery, the orange curve indicates the effect of gravitomagnetic tides from both the $m=1$ mode resonances and the adiabatic effects. Significant shifts in $\tilde{\mathrm{\Lambda }}$ and a peaked shape of the distribution of $\delta \tilde{\mathrm{\Lambda }}$ are clearly visible in this case. This is also illustrated by the two-dimensional representation of the error ellipses in the lower left panel.

Figure 8

Effects of various gravitomagnetic contributions on the parameter recovery for aligned spins. The results are for the systems with SNR 1800 and spins of $\chi =0.005$ (upper) and $\chi =0.01$ (lower). Green curves correspond to recovering with the same full model as used for the injection, blue curves include only the mode resonances, while orange curves indicate the adiabatic effects. We see that the conclusions about the impact of the resonance and adiabatic effect is opposite for the lower and higher spins: for low spins, adiabatic effects are most important for reducing the bias in $\tilde{\mathrm{\Lambda }}$, while resonances give the dominant contribution to the measurability of $\delta \tilde{\mathrm{\Lambda }}$. For high spins, the largest reduction in the bias in $\tilde{\mathrm{\Lambda }}$ is due to the resonances, while the impact on $\delta \tilde{\mathrm{\Lambda }}$ is comparable between resonance and adiabatic effects.

Figure 9

Gravitomagnetic effects for spin orientations $\psi =\pi /3$ and magnitudes $\chi =0.005$ at SNR 1800. The blue curve corresponds to using the same waveform for injection and recovery. Comparing this with the orange curve indicates the changes due to gravitomagnetic tides from both the $m=2$ mode resonances and the adiabatic effects, which lead to a shift in the distribution of $\tilde{\mathrm{\Lambda }}$ and a more peaked shape of the $\delta \tilde{\mathrm{\Lambda }}$ posterior.

Figure 10

Gravitomagnetic effects for spin orientations $\psi =\pi /3$ and magnitudes ${\chi }_{1,2}=0.01$ at SNR 1800. The blue curve corresponds to using the same waveform for injection and recovery. Comparing this with the orange curve indicates the changes due to gravitomagnetic tides from both the $m=2$ mode resonances and the adiabatic effects, which lead to a substantial shift in the distribution of $\tilde{\mathrm{\Lambda }}$ and clear peak in the $\delta \tilde{\mathrm{\Lambda }}$ posterior.

Figure 11

Effects of various gravitomagnetic contributions on the parameter recovery for misaligned spins. The results are for the systems with SNR 1800 and spin orientations of $\psi =\pi /3$ with $\chi =0.005$ (upper) and $\chi =0.01$ (lower). Green curves correspond to recovering with the same full model as the injection, blue curves include only the mode resonances, while orange curves indicate the adiabatic effects. We see that in both cases the mode resonances play a larger role for reducing biases than the adiabatic effects.

Figure 12

Effect of the spin magnitude on inferred tidal parameters for aligned spins and SNR of 1800. The injection and recovery both use the same model ${\text{PNTidal}}_{\mathrm{asym}}^{\text{modes}}$ with corresponding spin magnitudes, as indicated in the legend. Increasing the spin magnitude has very little impact on the width of the posterior in $\tilde{\mathrm{\Lambda }}$ but significantly affects that of $\delta \tilde{\mathrm{\Lambda }}$, where a higher spin leads to tighter bounds.

Figure 13

Accumulation of information encoded in integrands $\mathrm{Abs}(\frac{\partial {\tilde{h}}^{\star }}{\partial {\theta }_i}\frac{\partial \tilde{h}}{\partial {\theta }_i}*\frac{1}{S_n(f)})$ (normalized to its maximum value) for ${\theta }_i=\tilde{\mathrm{\Lambda }}$ (left) and ${\theta }_i=\delta \tilde{\mathrm{\Lambda }}$ (right) as a function of frequency for the injected value of aligned spins $\{0.0,0.005,0.01\}$, $\tilde{\mathrm{\Lambda }}=519.38$, and $\delta \tilde{\mathrm{\Lambda }}=48.37$. SNR denotes the integrands $\mathrm{Abs}(\frac{{\tilde{h}}^{\star }\tilde{h}}{S_n(f)})$, Electric denotes only adiabatic gravitoelectric tidal contribution in (3.1a), and Mag. all denotes adiabatic and resonant gravitomagnetic tidal contribution in (3.1a).

Figure 14

Effect of the spin magnitude on inferred tidal parameters for inclined spins at 60° and SNR of 1800. The injection and recovery both use the full model ${\text{PNTidal}}_{\mathrm{asym}}^{\text{modes}}$ with varying spin magnitudes as indicated in the legend. Increasing the spin magnitude from a finite value to a higher one has very little impact on the width of the posteriors in this case.

Figure 15

Bayesian parameter estimation results for systems with SNR 400 for different spins. The injection and recovery both use the model ${\text{PNTidal}}_{\mathrm{asym}}^{\text{modes}}$ with the corresponding spin magnitude indicated in the legend. Left: aligned spins. Right: spin inclinations of 60°. Same as Figs. 12 and 14 except for lower SNR.