Relativistic tidal properties of neutron stars

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 G\mu_\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 G\sigma_\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 G\mu_\ell/R^{2\ell+1}, G\sigma_\l/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'scompactness'' c\equiv GM/(c_0^2 R) (where we indicate by c_0 the speed of light). In particular, G\mu_\l/R^{2\l+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 \mu_\ell and \sigma_\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.

• The paper demonstrates that gravito-electric tidal coefficients significantly decrease with higher neutron star compactness, highlighting a relativistic quenching effect.
• The paper reveals that gravito-magnetic responses peak at intermediate compactness and become exceptionally small, complicating their detection in gravitational wave signals.
• The paper finds that shape Love numbers remain finite even at the black-hole compactness limit, offering critical insights for modeling neutron star mergers in gravitational wave astronomy.

Overview of the Paper on the Relativistic Tidal Properties of Neutron Stars

The paper, authored by Damour and Nagar, presents a detailed study of the relativistic tidal properties of neutron stars. It investigates how neutron stars, as compact astrophysical objects, respond to external tidal fields in a relativistic framework. The study is particularly relevant given the role of neutron stars in gravitational wave astronomy, especially in binary systems where tidal interactions significantly affect the emitted gravitational waveforms.

The authors classify the tidal responses of neutron stars into three types:

1. Gravito-electric responses characterized by a tidal coefficient $\mu_\ell$, measuring the mass multipolar moment caused by external gravito-electric fields.
2. Gravito-magnetic responses characterized by a tidal coefficient $\sigma_\ell$, measuring the spin multipolar moment caused by external gravito-magnetic fields.
3. "Shape" Love numbers, denoted as $h_\ell$, which quantify the deformation of the star's surface shape under external gravito-electric fields.

Methodological Approach

The paper employs analytical and numerical methods to explore how these tidal coefficients depend on the neutron star’s compactness, denoted by $c = GM/Rc_0^2$. This compactness parameter fundamentally influences tidal responses, especially near the black-hole limit of compactness $c^{\rm BH} = 1/2$.

For examining these tidal properties, different equations of state (EOS) for neutron-star matter are considered. These include both polytropic models and more realistic EOS like FPS and SLy. The insights gained from these models allow the authors to understand how variations in EOS and star compactness affect the values of $\mu_\ell$, $\sigma_\ell$, and $h_\ell$.

Key Findings

1. Electric-Type Tidal Response ($\mu_\ell$): The study reveals a strong dependence of the gravito-electric tidal Love numbers $k_\ell \sim G\mu_\ell/R^{2\ell+1}$ on the compactness of the neutron star. Notably, these coefficients decrease as the compactness increases, highlighting a significant quenching effect due to relativistic gravity. This dependency is partially attributed to the no-hair property of black holes, where tidal deformations vanish in the black-hole limit.
2. Magnetic-Type Tidal Response ($\sigma_\ell$): The gravito-magnetic coefficients show a more complex behavior. They tend to zero in both the Newtonian and black-hole limits, exhibiting a peak value at an intermediate compactness. Interestingly, $\sigma_\ell$ is negative and small, presenting challenges for measurability in gravitational wave analyses.
3. Shape Love Numbers ($h_\ell$): Unlike $\mu_\ell$ and $\sigma_\ell$, the shape Love numbers do not vanish in the black-hole limit. Instead, they approach finite values determined by recent studies on black hole polarizability. The study finds a marked reduction in $h_\ell$ from their Newtonian values, revealing a substantial quenching with increased compactness, similar to $\mu_\ell$.

Implications and Future Directions

The paper's findings have significant implications for gravitational wave astronomy and the modeling of neutron star mergers. Understanding the tidal properties of neutron stars helps in accurately predicting gravitational waveforms, crucial for inferring the internal structure and EOS of neutron stars from observations.

The study also sets the groundwork for future investigations into incorporating tidal effects within the Effective One Body (EOB) framework and assessing their impact on gravitational wave signal detectability in advanced interferometers. Such explorations are poised to deepen our understanding of neutron stars and possibly uncover novel facets of strong-field relativistic gravitation.

In summary, this comprehensive analysis of the tidal properties of neutron stars contributes valuable insights into the complex interplay between compactness, equation of state, and tidal interactions, enhancing our understanding of these celestial objects in the relativistic regime.