The Good, the Bad, and the Subtle: Relativistic mode sums for neutron-star tidal response

TL;DR

This paper develops a practical relativistic mode-sum approach to neutron-star tidal responses, demonstrating robustness of the dominant f-mode contribution and clarifying limitations due to gauge ambiguities and convergence issues.

Contribution

It introduces a relativistic implementation of mode-sum tidal response for non-rotating neutron stars in Regge-Wheeler gauge, highlighting its practical utility and limitations.

Findings

The dominant f-mode contribution is accurate within ~3% across various equations of state.

The mode sum truncated at order ω² does not strictly converge to the direct matching solution.

The interior tidal field is not unique, but this ambiguity has limited impact on the f-mode response.

Abstract

Time-dependent tidal interactions during the late inspiral of binary neutron stars encode valuable information about neutron-star structure, but systematically extending the familiar Newtonian mode-sum picture into full general relativity is nontrivial. In this paper, we develop a practical relativistic implementation of mode-sum tidal response for non-rotating neutron stars in Regge-Wheeler gauge. Using near-zone boundary conditions, we systematically define the interior tidal field, the relativistic overlap integrals, and the corresponding mode amplitudes. The good is that the dominant f-mode contribution is remarkably robust, reproducing the direct matching calculation to within $\sim 3$\% across the equations of state we consider. The bad is that the operator governing mode inner product is not positive definite on the full Regge-Wheeler-gauge function space, so the relativistic mode sum truncated at ${\cal{O}}(\omega^2)$ is not expected to strictly converge to the direct matching solution. The subtle is that the tidal field inside the star is not unique, although this ambiguity has only a limited impact on the dominant f-mode response for the classes of extensions studied here. Our results establish the practical utility of relativistic mode-sum approximations, while making clear that their predictive power comes from a controlled low-mode description, rather than from a formally convergent strong-field expansion.