Universal Relations with Dynamical Tides

TL;DR

The paper develops a relativistic framework for dynamical tides in neutron stars by expanding the frequency-dependent tidal response at small $\omega$, extracting the leading static term $\Lambda^{(0)}$ and the first dynamical correction $\Lambda^{(2)}$. It identifies two quasi-universal relations: (i) $\ln \Lambda^{(2)}$ versus $\ln \Lambda^{(0)}$ with $\lesssim 5\%$ EOS variability, and (ii) $M\omega_*$ versus $\ln \Lambda^{(0)}$ with $\lesssim 2.5\%$ variability, enabling dynamical tides to be modeled with essentially one independent parameter. The work compares Taylor and EoM approximations to the full dynamical tide and finds that both capture key frequency-dependent features up to about $f_{\rm GW} \sim 1$ kHz, with Taylor generally outperforming EoM at higher frequencies and for more compact stars. These universal relations simplify incorporating dynamical tidal effects into GW waveforms and offer a route for EOS-insensitive tests of gravity, while acknowledging caveats from neglected resonances and spin.

Abstract

Observations of neutron stars and the precise measurement of their macroscopic properties have provided valuable insights into fundamental physics, both by constraining the behavior of nuclear matter under extreme conditions and by enabling tests of general relativity in the strong-field regime. In this context, equation-of-state-insensitive or ``quasi-universal'' relations between key observables, such as the compactness, dimensionless static tidal deformability, and moment of inertia, play a crucial role in connecting different measurable observables while minimizing uncertainties due to the yet unknown equation of state. In this work, we identify new quasi-universal relations between the static, dimensionless tidal deformability ($Λ^{(0)}$) and its leading-order dynamical correction ($Λ^{(2)}$), as well as a combination of these parameters ($\sqrt{Λ^{(0)}/Λ^{(2)}} \equiv Mω_*$), obtained from the small-frequency expansion of the relativistic tidal response. We test these relations across a representative set of 63 equations of state, finding that the equation-of-state dependence does not exceed $\sim 5\%$ for the $Λ^{(0)}$-$Λ^{(2)}$ relation and $\sim 2.5\%$ for the $Λ^{(0)}$-$Mω_*$ relation. This indicates a high degree of universality and offers a simplified framework for incorporating dynamical tidal effects into gravitational-wave modeling. Furthermore, we compare the fully dynamical tidal response against different recent strategies (a Taylor expansion and an effective-one-mode approximation) to model the dynamical tide. We find that both models are capable of capturing the frequency-dependent behavior of the fully dynamical tidal deformability, with the Taylor expansion outperforming the effective-one-mode approximation in most of the parameter space.

Figures (6)

• Figure 1: Fractional difference between the dynamical tide and the static tide for different values of the compactness and the GW frequency. As frequency increases, so does the contribution of the dynamical part.
• Figure 2: Frequency-dependent tidal deformability for the SLy EOS. In both panels, the fully dynamical tide is represented by the solid blue line; the series approximation is represented by the green line with square markers, and the effective-one-mode approximation is represented by the orange line with diamond markers. The left panel shows the tidal deformability for $f_\text{GW}=200$ Hz while the right panel shows it for $f_\text{GW}=1300$ Hz.
• Figure 3: Percent fractional difference between the fully dynamical tide and the second-order small-frequency expansion (left) or the effective-one-mode approximation (right), as a function of compactness and GW frequency. As the frequency increases, the Taylor series expansion gradually loses accuracy, with deviations remaining below $\sim 25\%$ across most of the parameter space. Only at high frequencies, combined with very small compactness, does the error exceed this level. On the other hand, the effective-one-mode approximation remains accurate at low frequencies, with errors below $5\%$ for $\omega \lesssim 800\,\mathrm{Hz}$. At higher frequencies, its accuracy degrades, and the fractional difference exceeds $25\%$ for all compactness once the frequency rises above $1400\,\mathrm{Hz}$.
• Figure 4: Mass-Radius curves for the set of EOS considered in this paper. Each curve corresponds to a sequence of NSs for a particular EOS. Along each curve, the central pressure increases until it reaches the maximum mass allowed for the particular EOS. Observe that the various EOSs lead to wide changes in the mass-radius curves, although they all pass observational constraints to 90% confidence.
• Figure 5: Relation between the dimensionless static tidal deformability $\Lambda^{(0)}$ and its second-order frequency correction $\Lambda^{(2)}$ for the complete set of EOS. Compactness varies along each curve from the maximum allowed mass (left corner) down to $C=0.12$ (right corner). Top: Relation $\Lambda^{(2)}$--$\Lambda^{(0)}$ in log-log scale. Here, solid lines correspond to the tabulated EOS, dash-dotted lines to the PP EOS set, and the thick dashed line to the fit relation \ref{['eq:lambda_fit']}. Bottom: Fractional deviation from the best fit, expressed in percent. Observe that the maximum relative difference to the fit is $\sim 5\%$.