Dynamical tidal response of neutron stars: From effective field theory to gravitational waveforms

We investigate the fully relativistic dynamical tidal response of neutron stars up to second order in the frequency. Combining the worldline effective field theory for extended gravitating bodies with perturbation theory of relativistic stellar models, we derive the tidal deformation induced by an external time-dependent field, including a universal logarithmic running term. In the effective theory, we work in dimensional regularization and, through a consistent matching procedure, obtain for the first time the complete leading-order dynamical tidal corrections to both the conservative dynamics and the gravitational-wave signal of compact binaries, including the scheme-dependent finite terms in addition to the running. We show that, in the relativistic regime, dynamical effects cannot be fully captured by mode excitations alone. The magnitude of the additional contribution depends on the stellar compactness, the equation of state, and the running term. Dynamical Love numbers are significantly enhanced with respect to their static counterparts for relatively small compactness. As a result, although they formally enter the gravitational-wave phase at 8th post-Newtonian order, dynamical tidal effects yield a non-negligible contribution during the late inspiral. Using a Fisher-matrix analysis, we show that third-generation detectors such as the Einstein Telescope could measure dynamical Love numbers for a range of neutron-star masses and equations of state. Conversely, neglecting these effects can lead to significant biases in the inference of static Love numbers, and hence on the nuclear equation of state. Our results highlight the importance of dynamical tidal effects for high-precision gravitational-wave modeling with future detectors.

In this work we address the following question: How does the frequency dependence of the tidal response of relativistic neutron stars affect the dynamics of compact binaries and their GW emission? While previous studies established the qualitative importance of dynamical tides, a complete and systematic treatment connecting relativistic stellar perturbation theory, EFT response functions, binary dynamics, all the way down to GW observables is still lacking. This is particularly relevant in the presence of running terms, as is the case for dynamical tidal effects in compact objects, and to avoid ambiguities that inevitably plague perturbation-theory calculations, if the latter are not properly matched to GW observables.

In this work we complete this program. First, we compute the frequency-dependent tidal response of neutron stars by directly solving the relativistic perturbation equations, both perturbatively and nonperturbatively in frequency, without relying on a mode decomposition. Second, building on Ref. [50], we match the resulting response to the worldline EFT at loop order, thereby obtaining for the first time the dynamical Love numbers including both the logarithmic running and the scheme-dependent finite contributions in dimensional regularization. Crucially, matching the EFT to the underlying general-relativistic neutron-star dynamics in order to obtain the complete Love-number couplings cannot be achieved at tree level within the EFT. Instead, it requires the evaluation of higher-loop worldline diagrams at the classical level. These diagrams are formally divergent and therefore require regularization and renormalization, including the introduction of a subtraction prescription and an associated renormalization scale—a structure that closely mirrors the renormalization procedure and renormalization-group running familiar from quantum field theory. Matching the results of stellar perturbation theory to the EFT couplings enables the construction of observable quantities that are independent of the choice of coordinates and perturbation variables.

We further compare the full relativistic response with resonant models based on the dominant f-mode, quantifying the regime of validity of these approximations in agreement with recent results. Finally, after matching the stellar-perturbation theory response with the EFT action, we derive the complete leading-order dynamical tidal corrections to the conservative dynamics and GW signal of compact binaries. Since the tidal contributions to the PN waveform can themselves be derived from the EFT action, fixing the split between the tidal field and the response through matching to the EFT guarantees that the coefficients we compute are precisely those entering in the waveform. We then assess their detectability with current and future GW detectors for realistic neutron-star EoS. Interestingly, we find that—due to its strong enhancement at moderate compactness—the 8PN contribution coming from the dynamical Love number is relevant for parameter estimation with third-generation interferometers such as the Einstein Telescope (ET).

The dynamical tidal response coupling also exhibits a universal logarithmic running, in addition to a set of scheme-dependent finite terms, which are unambiguously fixed by the EFT matching and which we determine here for the first time using dimensional regularization. We have shown that, for a fixed running parameter, approximate EoS-insensitive relations exist between Λ0 and Λ2, analogous to the I-Love-Q universality, while taking motivated values of the running into account, the f-mode still provides an accurate single-mode approximation to the dynamical response throughout the late inspiral. Using these results, we derived the complete leading-order 8PN tidal phase correction to the gravitational waveform of a binary neutron-star inspiral, including both the EoS-dependent constant term and the logarithmic running. While the logarithmic term is qualitatively distinctive—in principle allowing one to disentangle the dynamical tide from unknown point-particle contributions at the same PN order—it is numerically subleading during the late inspiral. The waveform contribution associated with the dynamical Love number is also quantitatively comparable to that arising from the quadratic Love numbers. Both effects enter the GW phase at 8PN order and are enhanced in the low-compactness regime, scaling as ∼ C −8.

Our Fisher-matrix analysis shows that ET could measure the 8PN tidal contribution below the 100% level for mNS ≲ 1.3 M⊙ and mNS ≲ 1.8 M⊙ for the softest and stiffest EoS, respectively. In contrast, LVK O4 cannot constrain the dynamical tide for any of the configurations considered. The mismatch analysis reveals that omitting the 8PN dynamical tide introduces systematic biases in the inference of the static Love number—and hence the extracted nuclear EoS—for mNS ≲ 1.2 M⊙ and soft EoS, and larger masses for the stiffer ones, at ET sensitivity. This motivates including the full dynamical tidal phase in next-generation waveform models. In summary, our analysis shows that the dynamical Love number is potentially measurable with third-generation detectors for a range of neutron star masses and EoS, and that neglecting it can introduce systematic biases in the inference of nuclear-matter properties. The degree of observability depends sensitively on the EoS, with stiffer EoS yielding both a larger Λ8PN and a larger mismatch. These findings motivate including the full dynamical tidal phase in next-generation GW data-analysis pipelines, and suggest that joint inference of Λ5PN and Λ8PN with ET could provide a new window into the frequency-dependent tidal response of dense nuclear matter.

To improve AI systems—specifically those operating in high-stakes scientific simulation, gravitational wave data analysis, and complex physical modeling—the following targeted improvements should be implemented:

1. Implement a Relativistic Running Parameterization Module for Time-Series Forecasting

The paper demonstrates that tidal Love numbers are not constants but undergo logarithmic running (scale dependence) due to gravitational nonlinearities.

• Current AI Limitation: Most physics-informed neural networks (PINNs) treat material properties or coupling constants as static hyperparameters or simple time-dependent functions.

• Improved Capability: An AI system equipped with this module would model physical constants as scale-dependent variables, specifically incorporating logarithmic running terms (e.g., terms proportional to the renormalization group flow). This would allow the AI to accurately predict phase evolutions in dynamical systems where the interaction strength changes as a function of the observation scale (distance/frequency), preventing catastrophic drift in long-term simulations of coalescing compact objects.

2. Integrate Nonperturbative-to-Perturbative Hybrid Architectures

The paper utilizes both nonperturbative solutions to Lindblom-Detweiler equations and perturbative expansions in frequency to match Effective Field Theory (EFT) results.

• Current AI Limitation: Surrogate models often rely on either pure regression (which fails at resonances) or pure differential equation solvers (which are computationally expensive).

• Improved Capability: An AI architecture could use a Switching Loss Function that transitions between a perturbative regime (for low-frequency, stable states) and a nonperturbative solver (for high-frequency, resonant states). This would allow the AI to maintain high fidelity during critical phase transitions or resonance events in complex fluid dynamics or plasma physics without the computational cost of continuous full-system simulation.

3. Develop Scheme-Independent Objective Functions for Scientific Discovery

The paper highlights that tidal couplings are scheme-dependent and require rigorous matching to ensure coordinate independence and physical observability.

• Current AI Limitation: AI models trained on raw simulation data often learn coordinate artifacts—patterns that exist in the simulation's mathematical representation but not in the underlying physics.

• Improved Capability: By incorporating the paper’s matching procedure into the loss function, an AI can be forced to optimize for observable quantities (like the waveform phase) rather than intermediate, gauge-dependent variables. This ensures that scientific discoveries made by the AI are physically robust and not artifacts of the specific numerical grid or coordinate system used in training.

4. Enhance Sensitivity via 8PN-Aware Signal Detection Algorithms

The paper proves that dynamical tidal effects, despite appearing at a high Post-Newtonian (8PN) order, are significantly enhanced by compactness scaling and can be measured by next-generation detectors like the Einstein Telescope.

• Current AI Limitation: Deep learning filters for signal detection often prioritize the highest energy/amplitude components (the 5PN static term), treating higher-order corrections as noise.

• Improved Capability: An AI detector trained with Compactness-Scaled Attention Mechanisms would specifically weigh high-order, low-amplitude frequency corrections that scale with inverse compactness. This would allow the AI to detect faint but structurally critical signals in extremely noisy environments, significantly increasing the detection range for specialized astrophysical events (e.g., low-mass neutron star mergers).