Impact of the neutron star crust on the tidal polarizability

Background: The first direct detection of a binary neutron star merger by the LIGO Scientific Collaboration and the Virgo Collaboration (LIGO-Virgo) [Phys. Rev. Lett. 119, 161101 (2017)] has opened the brand new era of multimessenger astronomy. This historic detection has been instrumental in providing initial constraints on the tidal polarizability (or deformability) of neutron stars. In turn, the tidal polarizability—an observable highly sensitive to stellar compactness—has been used to impose limits on stellar radii and ultimately on the underlying equation of state (EOS).

Purpose: Besides its strong dependence on the stellar compactness, the tidal polarizability is also sensitive to the second tidal Love number $k_2$. It is the main purpose of this paper to perform a detailed study of $k_2$ which, for a given compactness parameter, encodes the entire sensitivity of the tidal polarizability to the underlying equation of state. In particular, we examine the important role that the crustal component of the EOS plays in the determination of $k_2$.

Methods: A set of realistic models of the equation of state that yield an accurate description of the properties of finite nuclei support neutron stars of two solar masses and provide a Lorentz covariant extrapolation to dense matter are used to solve both the Tolman-Oppenheimer-Volkoff and the differential equations for the induced quadrupole gravitational field from which $k_2$ is extracted.

Results: Given that the tidal polarizability scales as the fifth power of the compactness parameter, a universal relation exists among the tidal polarizability and the compactness parameter that is highly insensitive to the underlying equation of state. Thus, besides an extraction of the tidal polarizabilities, a measurement of the individual stellar masses is also required to impact the mass-radius relation. However, we observe a strong sensitivity of the second Love number to the underlying equation of state—particularly to the contribution from the inner crust.

Conclusions: Although by itself the tidal polarizability cannot contribute to the determination of the mass-radius relation, future detections of binary neutron star mergers by the LIGO-Virgo Collaborations during the third observing run and beyond are poised to provide significant constraints on both the tidal polarizabilities and the masses of the individual stars, and thus ultimately on the mass-radius relation. Yet, subleading corrections to the tidal polarizability are encoded in the second Love number $k_2$, which displays a large sensitivity to the entire—crust-plus-core—equation of state.

Figure 1

Neutron star matter equation of state as predicted by the relativistic density functional FSUGarnet [43]. The features that are sensitive to the choice of density functional are the crust-core transition pressure and the EOS in the entire uniform liquid core.

Figure 2

(a) The $y\left(r\right)$ profile as defined in Eq. (12) for a variety of neutron star masses as predicted by FSUGarnet [43]. The Love number is sensitive to $y_{\mathrm{R}}$, namely, the value of $y\left(r\right)$ at the surface of the star. (b) Same as in (a) but now exclusively for a $1.4\text{−}{M}_{\odot }$ neutron star with special emphasis on the contribution to $y\left(r\right)$ from various regions of the star.

Figure 3

The $y\left(r\right)$ profile as defined in Eq. (12) for a $1.4\text{−}{M}_{\odot }$ neutron star as predicted by a collection of ten relativistic models with different choices for the density dependence of the symmetry energy. The inset shows a correlation between $y_{\mathrm{R}}=y\left(R\right)$ and the neutron-skin thickness of ${}^{208}\mathrm{Pb}$ that is used as proxy for the slope of the symmetry energy $L$.

Figure 4

(a) Mass-radius relationship predicted by the ten models of the equation of state discussed in the text. Mass constraints from Refs. [85, 86] are indicated with a combined uncertainty bar and the radius constraint by an arrow as in Ref. [7]. The excluded causality region was adopted from Ref. [87]. (b) The dimensionless tidal polarizability as a function of both the stellar mass and the radius. The arrows indicate constraints inferred from the $\mathrm{\Lambda }\le 800$ limit for a $1.4\text{−}{M}_{\odot }$ neutron star [5]. In turn, the colored circles indicate the corresponding predictions for the location of a $1.4\text{−}{M}_{\odot }$ neutron star.

Figure 5

(a) The second tidal Love number $k_2$ as a function of the compactness parameter $\xi$ as predicted by the ten models of the equation of state used in the text. Although $k_2$ depends on $\xi$, the failure of all predictions to collapse into a single curve is due to its dependence on $y_{\mathrm{R}}$, which, in turn, depends on the entire EOS. (b) The same as (a) but now for the dimensionless tidal polarizability $\mathrm{\Lambda }$. The collapse of all ten predictions into a universal curve is due to the strong sensitivity of $\mathrm{\Lambda }$ to the compactness parameter $\xi$.