Tidal deformations of general-relativistic multifluid compact stars

Abstract: Over the past decade, gravitational-wave astronomy has opened a new window onto the extreme states of matter inside compact stars. At some point during the inspiral of a binary system, each star starts to experience adiabatic tides, characterized by tidal deformabilities. The dominant tidal deformability, first measured with the GW170817 event, has already constrained the dense-matter equation of state. With the advent of third-generation detectors, tidal deformabilities are expected to be inferred with much higher precision, potentially revealing subleading tidal contributions. This motivates the development of more accurate compact-star models that incorporate richer microphysics. With this in mind, we move beyond the commonly adopted perfect-fluid approximation and model compact stars through a multifluid framework. In this work, we present the fully general-relativistic description of adiabatic tidal deformations of compact stars composed of an arbitrary number of interacting fluids, using Carter's multifluid variational formalism. A distinctive feature of this approach is the presence of nondissipative mutual entrainment between fluid species. We derive the hydrostatic equilibrium equations for multifluid configurations, along with the perturbed equations governing stationary gravitoelectric and gravitomagnetic tidal responses of arbitrary order. We then investigate how entrainment modifies the corresponding tidal deformabilities. Using an analytical representation of the multifluid equation of state, we show that entrainment leaves adiabatic tidal responses unchanged and therefore produces no measurable effect on the gravitational-wave signal emitted during the inspiral long before the excitation of internal mode resonances. We subsequently discuss two specific applications: superfluid neutron stars and dark matter admixed compact stars.

• The paper rigorously demonstrates that adiabatic tidal Love numbers remain unaffected by nondissipative mutual entrainment effects in multifluid compact stars.
• It employs Carter’s variational multifluid framework to derive generalized Tolman-Oppenheimer-Volkoff equations and stationary tidal perturbation equations.
• The study supports using single-fluid models in gravitational-wave analyses since only the static energy density, not the entrainment, governs tidal responses.

Tidal Deformations in General-Relativistic Multifluid Compact Stars: A Formal Analysis

Introduction and Motivation

The study of tidal interactions in compact stars, especially in the context of gravitational-wave (GW) astronomy, has led to significant constraints on the equation of state (EOS) of dense matter. While the majority of theoretical models have relied on a single barotropic perfect-fluid approximation, physical conditions anticipated in neutron stars, strange stars, and exotic compact objects motivate the use of multifluid models that can encapsulate phenomena such as superfluidity, superconductivity, and dark matter admixture. This paper advances the field by constructing a fully covariant, general-relativistic formalism for the tidal deformations of compact stars composed of an arbitrary number of interacting fluids, with particular attention to the effects of mutual nondissipative entrainment within Carter’s variational multifluid framework (2603.29726).

General-Relativistic Multifluid Formalism

The formalism is centered on Carter’s variational multifluid dynamics, parameterized by a master function $\Lambda(n^\mu_{\mathrm{x}},g_{\mu\nu})$ that encapsulates all microstructural information, including cross-species couplings through terms depending on $$n^2_{\mathrm{xy}} = - n^\mu_{\mathrm{x}} n^\nu_{\mathrm{y}} g_{\mu\nu}$$. The action principle is constrained by requiring each fluid flux to be conserved. The resulting field equations comprise the Einstein field equations for the metric and a set of relativistic Euler equations for each fluid component, incorporating mutual entrainment via non-diagonal terms in $$\mathcal{A}_{\mathrm{xy}} = -\partial \Lambda/\partial n^2_{\mathrm{xy}}$$. Entrainment manifests through linear combinations of momenta and velocities across species. The formalism employs both Eulerian and Lagrangian variations, which are essential for handling perturbations in the tidal problem.

For equilibrium configurations, the multifluid generalization of the Tolman-Oppenheimer-Volkoff (TOV) equations is derived. Hydrostatic equilibrium is formulated as a system of $N+2$ coupled ordinary differential equations for the metric functions and the densities of $N$ fluid species, closed by the specification of the master function.

Stationary Tidal Perturbations and Love Numbers

The external tidal field from a binary companion during the inspiral phase is modeled as a stationary, adiabatic perturbation. Via multipolar decomposition, the problem naturally separates into polar (gravitoelectric) and axial (gravitomagnetic) sectors, each governed by a system of coupled linearized Einstein and multifluid Euler equations.

For the polar sector, a master equation for the metric perturbation $H_\ell$ is derived in terms of the background fluid and metric profiles, with a complex coupling encapsulated by a generalized functional $\mathcal{J}_N$ that depends on the multifluid configuration. The axial sector involves similar analysis, including careful treatment of static versus irrotational fluids in the context of gravitomagnetic responses.

The connection to observables is made via matching interior and exterior solutions at the stellar surface, leading to explicit expressions for the dimensionless electric ($k_\ell$) and magnetic ($j_\ell$) Love numbers for arbitrary multipoles in terms of the surface values of appropriately defined radial functions. These Love numbers directly enter GW waveform models and are key to EOS inference from astrophysical observations.

Analytical Structure and the Role of Entrainment

Crucially, using an analytic expansion of the master function,

$$\Lambda(n^2_{\mathrm{xy}}) = \mathcal{X}_0 + \frac{1}{2} \sum_{k=1}^{\infty} \sum_{\mathrm{z},\mathrm{w}} \mathcal{X}_k^{\mathrm{zw}} \left(n^2_{\mathrm{zw}} - n_{\mathrm{z}} n_{\mathrm{w}}\right)^k,$$

the analysis isolates the static energy density from entrainment contributions. It is rigorously demonstrated that all thermodynamic quantities relevant for tidal deformations are determined solely by the static part $$n^2_{\mathrm{xy}} = - n^\mu_{\mathrm{x}} n^\nu_{\mathrm{y}} g_{\mu\nu}$$0 (equivalently, the static EOS), and not by the entrainment coefficients. In other words, adiabatic tidal deformations—both gravitoelectric and gravitomagnetic, and at all multipole orders—are completely insensitive to the presence of nondissipative mutual entrainment.

This central result is obtained by reduction of the background hydrostatic and perturbation equations, showing that the effective variables entering the Love number equations can be written in terms of static energy density and pressure, with on-shell barotropicity emerging due to relations among the chemical potentials of the various fluid species.

Specific Applications

Superfluid Neutron Stars

The relevance of this result is made explicit in the context of superfluid neutron star models, which have historically been modeled using two-fluid (superfluid neutrons plus normal protons and electrons) formalisms. The numerical and analytical debate in the literature regarding the impact of superfluidity on tidal deformabilities is resolved: for beta-equilibrated, cold superfluid neutron stars, mutual entrainment effects do not contribute to tidal Love numbers, and all information is encapsulated in the barotropic EOS $$n^2_{\mathrm{xy}} = - n^\mu_{\mathrm{x}} n^\nu_{\mathrm{y}} g_{\mu\nu}$$1. Deviations previously reported in some numerical studies are attributed to artifacts or misinterpretations rather than true physical effects.

Dark Matter Admixed Compact Stars

Similarly, in models involving baryonic and dark matter components, including those allowing for interspecies couplings, the derivation shows that only the static energy density controls tidal deformabilities. Entrainment between baryons and dark matter does not affect the adiabatic tidal responses, validating the common approximation of neglecting dark-baryon entrainment in tidal analyses of dark matter admixed objects.

Implications and Outlook

The formal demonstration that adiabatic tidal Love numbers are independent of multifluid entrainment has significant implications:

• Theoretically, it establishes a universality in the mapping between observable tidal deformations and the static EOS, greatly simplifying the interpretation of GW data from binary inspirals.
• Practically, it justifies the widespread use of single-fluid models (with an appropriate static EOS) in GW parameter estimation pipelines, even for objects expected to have rich internal multifluid structure, as long as adiabatic tides dominate.
• For scenarios involving dynamical tides, rotation, or non-stationary backgrounds, the arguments will not, in general, hold—pointing to future avenues for extending the analysis.

Potential research directions include incorporation of charge-carrying fluids, dissipation, or full dynamical coupling to mode resonances. Extending Carter's formalism to generic time-dependent perturbations and arbitrary $$n^2_{\mathrm{xy}} = - n^\mu_{\mathrm{x}} n^\nu_{\mathrm{y}} g_{\mu\nu}$$2-fluid mixtures remains an open computational and conceptual challenge.

Conclusion

This work rigorously establishes that the adiabatic tidal Love numbers of general-relativistic multifluid compact stars are unaffected by nondissipative mutual entrainment, whichever microphysical origin or number of fluids is considered. Both the structure of the perturbed equations and the relationship to observables ensure that only the static energy density (the barotropic EOS) controls the response in the adiabatic regime. Superfluidity, dark matter admixture, and related multifluid effects are therefore "invisible" to adiabatic tidal GW measurements during early binary inspiral. This provides a robust theoretical basis for interpreting current and upcoming high-precision GW observations with simplified EOS models, and identifies the physical regimes where multifluid effects may become relevant in future work (2603.29726).