Neutron Stars as Perfect Fluids: Extracting the Linearized Response Function

TL;DR

This work develops a covariant effective-field-theory framework to extract the general relativistic linear tidal response of a neutron star modeled as a barotropic perfect fluid. By matching bulk fluid dynamics in curved spacetime to a worldline description of a dynamical quadrupole, the authors derive a mode-based (mode-sum) representation of the stellar response, where each normal mode acts as a tidally driven oscillator with frequency $\omega_n$, normalization $\mathcal{N}_n$, and overlap $\mathcal{I}_n$. The formalism yields analytic expressions for dynamical tidal deformabilities $\lambda_n$ and Love numbers $k_n$ in terms of mode properties, and provides a systematic bridge between relativistic stellar perturbations and gravitational-wave observables. The framework clarifies the range of validity, emphasizes the role of the complete mode spectrum, and sets the stage for numerical implementation and extensions to more general equations of state and nonlinear regimes.

Abstract

We derive the general relativistic linear tidal response of a neutron star modeled as a barotropic perfect fluid. From the covariant fluid effective action, we linearize about equilibrium and obtain the action for fluid displacements coupled to metric perturbations. Splitting the latter into external and induced parts and integrating out the induced field yields a Hermitian operator and a discrete gapped spectrum of driven modes. Projecting the displacement onto this eigenbasis and integrating out the spatial dependence over the stellar radius reduces the dynamics to tidal-driven oscillators, with couplings set by relativistic inner products and overlap integrals. Matching to the quadrupolar worldline effective action gives a mode-sum response function and analytic dynamical tidal deformabilities from mode frequencies, normalizations, and overlaps.

Figures (2)

• Figure 1: Schematic illustration of the hierarchy of scales involved in the problem. On the right hand panel there is a well separated binary, where the point particle approximation is valid. In the middle panel we have zoomed in into the point-like particle, to find the star with radius $R_{\star}$. On the left panel, we have the hydrodynamical scale of the neutron perfect fluid.
• Figure 2: Feynman diagrams capturing the leading gravitational interaction involving fluid perturbations. The horizontal line denotes graviton exchange, the dashed vertical line the fluid displacement, and the solid vertical line an external point-mass source. Panel (a) shows the interaction between two fluid elements inside the star, while (b) shows the interaction between a fluid element and the external point particle.