Nonlinear hydrodynamics in spinning neutron stars: Theoretical universal relations and equilibrium solutions

Author(s)

Yu, Hang, Nicolini, Giorgio, Lau, Shu Yan, Kwon, K.J., Venumadhav, Tejaswi, Andersson, Nils, Pnigouras, Pantelis, Gittins, Fabian, Nanda, Amlan

Abstract

We study tides during the inspiral of a binary neutron star (BNS) system, including nonlinear hydrodynamical interactions. Using an affine approximation that treats the perturbed NS as an ellipsoid, we analytically derive coupling coefficients among the f-modes and the radial mode to the four-wave order (i.e., next-to-next-to-leading order) in the Hamiltonian, allowing for arbitrary rotation of the background star. Our model reveals a series of universal relations from first-principles arguments. Besides the well-known relations, we show that the three-wave (next-to-leading-order) interaction coefficients are fully determined by the properties of the linear tide. Therefore, they do not probe new physics of the NS. Nonetheless, not including the three-wave nonlinear tides can lead to significant systematic errors in the gravitational waveform. We support this claim via a hybrid approach that simultaneously captures mode resonances expected in Newtonian hydrodynamics and is consistent with relativistic calculations in the low-frequency expansion. The nonlinear tide in a single NS can cause a phase shift of around 1.7 radians accumulated up to merger compared to the linear tide model; for a binary of similar masses, the phase shift is approximately doubled. Our calculation extends to four-wave interactions, which, for a slowly spinning NS, provide only small corrections and are subdominant compared to the tidal back-reaction on the orbit. For a rapidly rotating NS, the nonlinear centrifugal drive of the f-mode and the four-wave anharmonicity provides a window to study the adiabatic exponent related to internal buoyancy that cannot be probed by the linear and three-wave tides in slowly spinning systems. The anharmonicity cannot lead to resonance locking of the f-mode.

Figures

Comparison of the calculated $\ddot{k}_{2A}$ (shown in gray) with prediction from the universal relation (shown in yellow). On the left, we show $\ddot{k}_{2A}$ as a function of the compactness $(M_A/R_A)$. Unlike previous empirical relations (\cite{Saes:25} in orange-dotted, \cite{Chan:14} in blue-dotted) that have limited range of validity, our theoretically informed relation holds for the entire range of compactness. There are theoretical uncertainties in the relativistic regime, leading to differences among predictions in the $M_A/R_A>0.1$ range. On the right, we show $\bar{\lambda}_{A}^{(2)}$ as a function of $\bar{\lambda}_A$. The approximate $\propto \lambda_A^{3/2}$ scaling can be analytically argued from Eqs. (\ref{eq:ddk2_vs_k2_wf}) and (\ref{eq:lambar2_vs_lambar}).
Caption Comparison of the calculated $\ddot{k}_{2A}$ (shown in gray) with prediction from the universal relation (shown in yellow). On the left, we show $\ddot{k}_{2A}$ as a function of the compactness $(M_A/R_A)$. Unlike previous empirical relations (\cite{Saes:25} in orange-dotted, \cite{Chan:14} in blue-dotted) that have limited range of validity, our theoretically informed relation holds for the entire range of compactness. There are theoretical uncertainties in the relativistic regime, leading to differences among predictions in the $M_A/R_A>0.1$ range. On the right, we show $\bar{\lambda}_{A}^{(2)}$ as a function of $\bar{\lambda}_A$. The approximate $\propto \lambda_A^{3/2}$ scaling can be analytically argued from Eqs. (\ref{eq:ddk2_vs_k2_wf}) and (\ref{eq:lambar2_vs_lambar}).
Left: Comparison of $p_{2A}$ obtained from \cite{Pitre:25} (solid-gray) and those predicted by the universal relations (Eq. \ref{eq:p2_vs_ddk2_GR}; dashed-yellow). Right: Comparison of the $\bar{p}_2-\bar{\lambda}_A$ quasi-universal fit, Eq. (\ref{eq:p2bar_vs_lambar}) with tracks computed for different polytropes.
Caption Left: Comparison of $p_{2A}$ obtained from \cite{Pitre:25} (solid-gray) and those predicted by the universal relations (Eq. \ref{eq:p2_vs_ddk2_GR}; dashed-yellow). Right: Comparison of the $\bar{p}_2-\bar{\lambda}_A$ quasi-universal fit, Eq. (\ref{eq:p2bar_vs_lambar}) with tracks computed for different polytropes.
Amplitude of the $l=m=2$ f-mode that dominates the tidal interaction. A non-spinning, Newtonian $\Gamma=\Gamma_{\rm ad}=2$ polytrope is assumed. Below 1000\,Hz, the four-wave approximation agrees well with numerical simulations. At higher frequencies, whether tidal back-reaction is included can have significant impacts on the mode dynamics (red-solid vs. red-dashed). This is mainly through the effective damping of the mode that depends on $\dot{\omega}$, which can be enhanced by an order of magnitude by the tide compared to the PP value when the f-mode approaches resonance. Near resonance, the equilibrium tide solution also tends to overestimate the mode amplitude.
Caption Amplitude of the $l=m=2$ f-mode that dominates the tidal interaction. A non-spinning, Newtonian $\Gamma=\Gamma_{\rm ad}=2$ polytrope is assumed. Below 1000\,Hz, the four-wave approximation agrees well with numerical simulations. At higher frequencies, whether tidal back-reaction is included can have significant impacts on the mode dynamics (red-solid vs. red-dashed). This is mainly through the effective damping of the mode that depends on $\dot{\omega}$, which can be enhanced by an order of magnitude by the tide compared to the PP value when the f-mode approaches resonance. Near resonance, the equilibrium tide solution also tends to overestimate the mode amplitude.
Similar to Fig. \ref{fig:b2_vs_f_N} but for a relativistic $\Gamma=\Gamma_{\rm ad}=2$ polytrope whose parameters are set following Sec. \ref{sec:calibrations}. As tidal coupling is weaker in a relativistic NS than in a Newtonian one, the three-wave (NLO) solution shown in gray lines is sufficient to match the numerical result in red lines.
Caption Similar to Fig. \ref{fig:b2_vs_f_N} but for a relativistic $\Gamma=\Gamma_{\rm ad}=2$ polytrope whose parameters are set following Sec. \ref{sec:calibrations}. As tidal coupling is weaker in a relativistic NS than in a Newtonian one, the three-wave (NLO) solution shown in gray lines is sufficient to match the numerical result in red lines.
Top: numerical solutions of the $l=m=2$ f-mode in a rapidly spinning NS with $\Omega_A/2\pi=-800\,{\rm Hz}$. A relativistic $\Gamma=2$ polytrope is assumed for the coupling coefficients. The gray line shows the mode solution including all nonlinear interactions. Its real and imaginary parts are shown respectively in the red-dashed and purple-dashed lines, indicating how the mode oscillates beyond resonance. The linear tide solution is represented by the yellow curve, which is qualitatively similar to the nonlinear case. The two vertical lines indicate the estimated f-mode resonance frequency from linear and three-wave approximations. Bottom: Evolution of the tidal spin $\chi_t$. In both the linear and nonlinear cases, the tidal spin evolves roughly linearly with frequency, which is due to the transient response of a driven harmonic oscillator, instead of nonlinear resonance locking.
Caption Top: numerical solutions of the $l=m=2$ f-mode in a rapidly spinning NS with $\Omega_A/2\pi=-800\,{\rm Hz}$. A relativistic $\Gamma=2$ polytrope is assumed for the coupling coefficients. The gray line shows the mode solution including all nonlinear interactions. Its real and imaginary parts are shown respectively in the red-dashed and purple-dashed lines, indicating how the mode oscillates beyond resonance. The linear tide solution is represented by the yellow curve, which is qualitatively similar to the nonlinear case. The two vertical lines indicate the estimated f-mode resonance frequency from linear and three-wave approximations. Bottom: Evolution of the tidal spin $\chi_t$. In both the linear and nonlinear cases, the tidal spin evolves roughly linearly with frequency, which is due to the transient response of a driven harmonic oscillator, instead of nonlinear resonance locking.
Tidal phase shifts for NS models with different $\Gamma_{\rm ad}$ but otherwise identical parameters. In the left panel, we evolve a rapidly spinning NS with $\Omega_A/2\pi=-800\,{\rm Hz}$. The system shown in the gray line is identical to the one shown in Fig. \ref{fig:no_res_lock} with the same color. The yellow line instead has $\Gamma_{\rm ad}=2.5$. The estimated times of resonance, including four-wave corrections, are shown in the vertical dotted lines. Right: for an NS with $\Omega_A/2\pi=-450\,{\rm Hz}$. Resonance of the f-mode does not occur until $r\simeq 2R_A$. The impact of $\Gamma_{\rm ad}$ is much smaller in such a system.
Caption Tidal phase shifts for NS models with different $\Gamma_{\rm ad}$ but otherwise identical parameters. In the left panel, we evolve a rapidly spinning NS with $\Omega_A/2\pi=-800\,{\rm Hz}$. The system shown in the gray line is identical to the one shown in Fig. \ref{fig:no_res_lock} with the same color. The yellow line instead has $\Gamma_{\rm ad}=2.5$. The estimated times of resonance, including four-wave corrections, are shown in the vertical dotted lines. Right: for an NS with $\Omega_A/2\pi=-450\,{\rm Hz}$. Resonance of the f-mode does not occur until $r\simeq 2R_A$. The impact of $\Gamma_{\rm ad}$ is much smaller in such a system.
Comparison between analytical and numerical calculations of the effective Love number. The background NS is the same as the one considered in Fig. \ref{fig:b2_vs_f_G}.
Caption Comparison between analytical and numerical calculations of the effective Love number. The background NS is the same as the one considered in Fig. \ref{fig:b2_vs_f_G}.
Comparison of non-PP energies driven by radial (gray) and tangential (i.e., tidal torque; yellow) interactions. The values are numerically extracted for a non-spinning $\Gamma=2$ relativistic polytrope. Tidal torque's contribution is initially negligible but becomes significant, even dominant, in the late inspiral.
Caption Comparison of non-PP energies driven by radial (gray) and tangential (i.e., tidal torque; yellow) interactions. The values are numerically extracted for a non-spinning $\Gamma=2$ relativistic polytrope. Tidal torque's contribution is initially negligible but becomes significant, even dominant, in the late inspiral.
GW phase shift for a relativistic $\Gamma=2$, non-spinning polytrope. The left panel shows the frequency-domain GW phase shift relative to a PP orbit. The red and purple lines show the phase shift with and without the nonlinear tide. The brown-dashed line shows the low-frequency limit including NLO nonlinear tide (Eqs. \ref{eq:phase_shift_LF} and \ref{eq:k_psi_eff}), which significantly underestimates the result. The right panel shows the time-domain GW phases (blue for PP, purple for linear tide, and red for nonlinear tide), which demonstrates that the actual effect of the nonlinear tide (red) is more significant than what the left panel shows. At the last time point where the system with the nonlinear tide merges ($t/M_A\simeq-160$), the nonlinear tide causes a phase shift of 0.22 rad relative to the linear tide system, same as the phase shift at the last frequency data point in the left up to a minus sign). However, the total GW phase accumulated to the merger ($r=2R_A$) is less by 0.81 rad due to the nonlinear tide.
Caption GW phase shift for a relativistic $\Gamma=2$, non-spinning polytrope. The left panel shows the frequency-domain GW phase shift relative to a PP orbit. The red and purple lines show the phase shift with and without the nonlinear tide. The brown-dashed line shows the low-frequency limit including NLO nonlinear tide (Eqs. \ref{eq:phase_shift_LF} and \ref{eq:k_psi_eff}), which significantly underestimates the result. The right panel shows the time-domain GW phases (blue for PP, purple for linear tide, and red for nonlinear tide), which demonstrates that the actual effect of the nonlinear tide (red) is more significant than what the left panel shows. At the last time point where the system with the nonlinear tide merges ($t/M_A\simeq-160$), the nonlinear tide causes a phase shift of 0.22 rad relative to the linear tide system, same as the phase shift at the last frequency data point in the left up to a minus sign). However, the total GW phase accumulated to the merger ($r=2R_A$) is less by 0.81 rad due to the nonlinear tide.
Similar to Fig. \ref{fig:phase_shift} but $M_A$ is now assumed to be described by the SLy equation of state with $(k_{2A}, \ddot{k}_{2A}, p_{2A})=(0.084, 0.060, 0.086)$. Compared to the linear tide prediction, the nonlinear tidal response of a single NS causes a phase shift of 0.57 rad measured at the time when the nonlinear system merges, and 1.7 rad when considering the total accumulated phase to $r=2R_A$.
Caption Similar to Fig. \ref{fig:phase_shift} but $M_A$ is now assumed to be described by the SLy equation of state with $(k_{2A}, \ddot{k}_{2A}, p_{2A})=(0.084, 0.060, 0.086)$. Compared to the linear tide prediction, the nonlinear tidal response of a single NS causes a phase shift of 0.57 rad measured at the time when the nonlinear system merges, and 1.7 rad when considering the total accumulated phase to $r=2R_A$.
GW phase shift caused by relativistic SLy NSs (parameters same as Fig. \ref{fig:phase_shift_SLy}) but with different spins (with colors gray, red, and yellow for positive, zero, and negative spins). Nonlinear tides are included in all models. The left panel shows the frequency-domain phase shift relative to the non-spinning system. The right panel shows the time-domain phase shift near the merger. For slowly rotating NSs, a small change in the spin rate of $|\Omega_A|/\omega_A\simeq0.05$ ($|\chi_A|\simeq 0.04$) leads to a total phase shift similar to that caused by the nonlinear tide.
Caption GW phase shift caused by relativistic SLy NSs (parameters same as Fig. \ref{fig:phase_shift_SLy}) but with different spins (with colors gray, red, and yellow for positive, zero, and negative spins). Nonlinear tides are included in all models. The left panel shows the frequency-domain phase shift relative to the non-spinning system. The right panel shows the time-domain phase shift near the merger. For slowly rotating NSs, a small change in the spin rate of $|\Omega_A|/\omega_A\simeq0.05$ ($|\chi_A|\simeq 0.04$) leads to a total phase shift similar to that caused by the nonlinear tide.
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