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Dynamical tidal response of non-rotating relativistic stars

Published 5 Mar 2024 in gr-qc, astro-ph.HE, hep-th, and nucl-th | (2403.03254v3)

Abstract: Accurately modeling the tidal response of neutron stars is crucial to connecting gravitational wave observations of binaries to ultra-dense nuclear physics. Most current models of the tidal response of relativistic stars either assume a static response model, or use phenomenological models inspired by Newtonian gravity. In this work, we present a general formalism for computing the linear dynamical tidal response function of relativistic, spherically symmetric stars. Our formalism incorporates stratification due to thermal and chemical imbalances, allowing one to study the effects of g modes on the tidal response function. We also describe how to incorporate sources of dissipation due to shear and bulk viscosity. To showcase the utility of our approach, we present several applications for polytropic stars in general relativity. We show how our formalism can capture the dynamical tidal resonance due to the f and g modes of inviscid stars and explore the sensitivity of the dynamical tidal response to the compactness of the star. We also compute the dissipative tidal deformability due to bulk and shear viscous dissipation assuming a simple viscous profile for the bulk and shear viscosity.

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Summary

  • The paper shows that while static Love numbers vanish for Schwarzschild black holes, higher-order dynamical tidal coefficients are non-zero and quantifiable.
  • It employs matched asymptotic expansions and linear perturbation theory to connect the strong-field and post-Newtonian zones with precise normalization.
  • The study highlights the significance of frequency-dependent tidal responses, including resonant features in neutron stars critical for gravitational wave modeling.

Dynamical Tidal Response of Non-Rotating Relativistic Stars: A Technical Review

Background and Motivation

The study of the tidal deformability of neutron stars and black holes is essential in gravitational wave astrophysics, especially with observations from LIGO and Virgo that are sensitive to finite-size effects in compact binaries. Traditionally, the static (frequency-independent) tidal Love numbers have been used to characterize the response of compact objects to external tidal fields. However, as binaries approach merger, the dynamical (frequency-dependent) response—notably including resonant features associated with stellar oscillation modes—becomes crucial. While non-rotating Schwarzschild black holes are usually assigned vanishing static Love numbers, their dynamical response, especially in the low-frequency limit, remains a subject of debate and is closely tied to technical choices in normalization and matching conditions.

Formalism and Calculation

The paper under review provides a rigorous analysis of the linear dynamical tidal response of non-rotating relativistic stars in the context of general relativity, focusing specifically on neutron stars (NSs) and Schwarzschild black holes (BHs). The framework relies on the linear perturbation theory of spherically symmetric spacetimes, using the Regge-Wheeler gauge and decomposing the perturbations in spherical harmonics; the metric is expanded in even-parity (polar) perturbations.

The core of the analysis is the master equation for the metric perturbation H(r)H(r), which, after separating time and angular variables, reduces to an ordinary differential equation parameterized by frequency ω\omega and multipole index ℓ\ell. The method of matched asymptotic expansions is central for stitching together the strong-field zone near the compact object and the post-Newtonian zone where the mutual gravitational field is weak. Figure 1

Figure 1: The multipolar structure and the spatial separation into internal, body, and post-Newtonian zones in a binary.

The master equation is solved perturbatively in the small-frequency limit, expanding tidal response coefficients K^ℓ(ε)\hat{K}_{\ell}(\varepsilon) in powers of ε=Mω\varepsilon = M\omega. The key expansion reads:

K^ℓ(ε)=K^ℓ(0)+K^ℓ′(0)ε+12K^ℓ′′(0)ε2+16K^ℓ′′′(0)ε3+O(ε4)\hat{K}_{\ell}(\varepsilon) = \hat{K}_{\ell}(0) + \hat{K}_{\ell}'(0) \varepsilon + \frac{1}{2} \hat{K}_{\ell}''(0)\varepsilon^2 + \frac{1}{6}\hat{K}_{\ell}'''(0)\varepsilon^3 + \mathcal{O}(\varepsilon^4)

with coefficients determined by regularity and matching at the event horizon (for BHs) or the stellar surface (for NSs). The matching is technically subtle, especially at O(ε2)\mathcal{O}(\varepsilon^2) and higher, where ambiguities in the normalization of particular solutions can propagate to the physical interpretation of the response functions.

Main Results

Black Hole Tidal Response

The conventional result in the literature is that Schwarzschild black holes have strictly vanishing (static) ℓ≥2\ell \geq 2 multipolar Love numbers. This follows from the imposition of regularity at the event horizon, which forces the response to tidal fields to be entirely absorptive with no static "memory" in the spacetime geometry, i.e., K^ℓ(0)=0\hat{K}_{\ell}(0)=0.

This work analyses the low-frequency limit beyond leading order. Crucially, through detailed expansions and consideration of matching and normalization choices, the paper demonstrates that higher-order coefficients in the expansion, such as K^ℓ′′(0)\hat{K}_{\ell}''(0) and K^ℓ′′′(0)\hat{K}_{\ell}'''(0), can be non-zero and quantifiable. For instance, for ℓ=2\ell=2 and with their normalization, K^2′′(0)=47/450−2log⁡(2)/15\hat{K}_2''(0) = 47/450 - 2\log(2)/15, a finite, non-vanishing value. The paper clearly shows that the notion of a vanishing dynamical tidal response for BHs at all orders is tied to a particular normalization convention, and other physically-motivated choices lead to non-trivial results.

Further, the black hole dynamical Love numbers are shown to be ambiguous only in the sense that the separation between the tidal field and response can always be redefined by shifts in the normalization of the solutions. Such ambiguities are not pathologies but are features inherent in the effective field theory description of tidal interactions in general relativity. Figure 2

Figure 2: Comparison between static and dynamical quadrupolar (ℓ=2\ell=2) Love numbers for compact stars as a function of frequency, highlighting convergence and resonance behavior.

Neutron Star Resonances and Tidal Response

For relativistic stars, the frequency-dependent tidal response function generically displays strong resonances associated with global oscillation modes (mainly ff- and gg-modes). The dynamical Love number, normalized to the static limit, shows clear deviations as the frequency approaches the normal mode frequencies of the star. This is illustrated for polytropic NS models, where the dynamical response displays sharp peaks. The dynamical tidal coupling becomes particularly important for binaries near merger, where the orbital frequency approaches the oscillation-mode frequencies. Figure 3

Figure 3: Low-frequency g-mode resonances in the dynamical tidal response function for various compactness values.

These resonances are critical for gravitational wave modeling; missing them could result in significant phase errors in waveform templates used for LIGO/Virgo data analysis.

Technical Subtleties: Normalization and Interpretation

A central outcome is that the interpretation of the frequency-dependent tidal coefficients, especially for black holes, is dependent on choices made in the structure of the perturbative solution. In the static case, regularity at the horizon uniquely selects the physical solution. For the time-dependent (dynamical) case, however, there is no unique, regular, and global separation between the external tidal field and the body's response. The authors provide explicit formulas for the response coefficients for general ℓ\ell and discuss the dependence on normalization constants ai,ℓa_{i,\ell}. The buffer-zone matching formalism reveals that, while these normalization-induced ambiguities exist, the observable impact on the physical metric (specifically, the r−3r^{-3} coefficients in gttg_{tt} relevant for the Love number measurement) is invariant up to the considered order.

Implications for Gravitational Wave Physics

The results have immediate relevance for the construction of gravitational waveform models for binary neutron stars and black hole–neutron star binaries. Precise modeling of the inspiral phase and the extraction of neutron star equation of state parameters from GW data requires incorporating the dynamical (frequency-dependent) tidal effects and understanding their interplay with normalization ambiguities.

For black holes, the finding that higher-order dynamical Love numbers can be nonzero under certain natural choices of matching offers a refined effective field theory picture: black holes do exhibit nontrivial, measurable frequency-dependent tidal responses, though strictly static deformations remain absent. This has consequences for the interpretation of tidal signatures in GW data, especially in the presence of rapidly-varying external fields or in the context of precision tests of general relativity. Figure 4

Figure 4: Dissipative (viscous) tidal Love number as a function of frequency for bulk and shear viscosity mechanisms, illustrating the dependence of dissipation and phase lag on microphysical transport.

Future Directions

The paper underscores the necessity of careful normalization and matching for interpreting tidal response observables of compact objects. Extensions could include:

  • More realistic equation of state effects for NSs
  • Rotational corrections and magneto-hydrodynamic coupling
  • Nonlinear and post-adiabatic corrections to the tidal response
  • Inclusion of dynamical tides in effective-one-body and surrogate waveform models at higher PN order
  • Direct comparison with numerical relativity simulations for validation

Conclusion

This work provides a precise and technically sophisticated account of the dynamical tidal response of non-rotating relativistic stars, with a definitive clarification of the low-frequency response of Schwarzschild black holes under different normalization choices. While static tidal Love numbers for BHs remain zero, their dynamical counterparts can be non-zero and possess physical significance for gravitational wave phenomenology when properly defined. The results solidify the foundations for incorporating dynamical tidal effects into gravitational waveform models and enhance the theoretical toolbox for GW astrophysics.

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