Relativistic tidal properties of neutron stars

TL;DR

This work provides a comprehensive relativistic analysis of neutron star tidal responses to external fields, introducing and computing electric-type $\mu_\ell$, magnetic-type $\sigma_\ell$, and shape $h_\ell$ tidal coefficients. Using stationary perturbation theory in the DSX framework and a interior–exterior matching procedure, the authors solve the relevant Master equations for even and odd parity, obtaining explicit relations between the interior structure (via the TOV background) and the external tidal fields. A key result is the strong, universal suppression of $k_\ell$ (and hence $G\mu_\ell/R^{2\ell+1}$) as compactness $c$ increases, driven by a factor $$(1-2c)^2$$ and consistent with BH no-hair properties; in contrast, $h_\ell$ remains finite and tends to BH values $h_2^{\rm BH}=1/4$, $h_3^{\rm BH}=1/20$ as $c\to1/2$. Across polytropic, incompressible, and realistic EOS, the study finds near-universal behavior in the leading quadrupolar electric response, offering potential to constrain NS compactness from gravitational-wave observations; magnetic-type responses are smaller and sign-defined, while shape deformations persist in the BH limit. These results refine the EFT/DSX description of tidal effects and set the stage for incorporating tides into the EOB framework for accurate GW waveform modeling.

Abstract

We study the various linear responses of neutron stars to external relativistic tidal fields. We focus on three different tidal responses, associated to three different tidal coefficients: (i) a gravito-electric-type coefficient Gμ_\ell=[length]^{2\ell+1} measuring the \ell^{th}-order mass multipolar moment GM_{a_1... a_\ell} induced in a star by an external \ell^{th}-order gravito-electric tidal field G_{a_1... a_\ell}; (ii) a gravito-magnetic-type coefficient Gσ_\ell=[length]^{2\ell+1} measuring the \ell^{th} spin multipole moment G S_{a_1... a_\ell} induced in a star by an external \ell^{th}-order gravito-magnetic tidal field H_{a_1... a_\ell}; and (iii) a dimensionless ``shape'' Love number h_\ell measuring the distortion of the shape of the surface of a star by an external \ell^{th}-order gravito-electric tidal field. All the dimensionless tidal coefficients Gμ_\ell/R^{2\ell+1}, Gσ_ł/R^{2\ell+1} and h_\ell (where R is the radius of the star) are found to have a strong sensitivity to the value of the star's ``compactness'' c\equiv GM/(c_0^2 R) (where we indicate by c_0 the speed of light). In particular, Gμ_ł/R^{2ł+1}\sim k_\ell is found to strongly decrease, as c increases, down to a zero value as c is formally extended to the ``black-hole (BH) limit'' c^{BH}=1/2. The shape Love number h_\ell is also found to significantly decrease as c increases, though it does not vanish in the formal limit c\to c^{BH}. The formal vanishing of μ_\ell and σ_\ell as c\to c^{BH} is a consequence of the no-hair properties of black holes; this suggests, but in no way proves, that the effective action describing the gravitational interactions of black holes may not need to be augmented by nonminimal worldline couplings.

Figures (10)

• Figure 1: Polytropic EOS: gravito-electric Love numbers $k_\ell$ (or apsidal constants) for $\ell=2,3,4$ versus compactness $c=M/R$. We use two different polytropic EOS's, either of the rest-mass type ($p=\kappa\mu^\gamma$; solid lines) or of the energy type ($p=\kappa e^\gamma$; dashed lines). For both EOS's we use $\gamma=2$. Note that the maximum compactness allowed by the $e$-polytrope is larger than that for the $\mu$-polytrope.
• Figure 2: Normalized Love numbers versus $c$ for a $\gamma=2$$\mu$-polytrope (points); and performance of the fitting with the template given by Eq. \ref{['eq:fit']} (dashed lines). The coefficients of the fit for each $\ell$ are listed in Table \ref{['tab:table1']}.
• Figure 3: Incompressible EOS: gravito-electric Love numbers $k_\ell$ (or apsidal constants) for $\ell=2,3,4$ versus compactness $c=M/R$.
• Figure 4: The gravito-electric Love numbers $k_\ell$ (or apsidal constants) for $\ell=2,3,4$ versus compactness $c=M/R$ for the two tabulated "realistic" equation of state FPS and SLy (left panel). Right panel: comparison between $k_2$ from various relativistic $\mu$-polytropes (with different $\gamma$) and the FPS and SLy realistic EOS's.
• Figure 5: The $j_2$, odd-parity, Love number for the ${\gamma}=2$, $\mu$-polytrope and for the incompressible EOS.