Impact of updated multipole Love numbers and $f$-Love universal relations in the context of binary neutron stars

Neutron star (NS) equation of state (EoS) insensitive relations or universal relations (UR) involving neutron star bulk properties play a crucial role in gravitational-wave astronomy. Considering a wide range of equations of state originating from (i) phenomenological relativistic mean field models, (ii) realistic EoS models based on different physical motivations, and (iii) polytropic EoSs described by spectral decomposition method, we update the EoS-insensitive relations involving NS tidal deformability (multipole Love relation) and the UR between $f$-mode frequency and tidal deformability ($f$-Love relation). We analyze the binary neutron star event GW170817 using the frequency domain TaylorF2 waveform model with updated universal relations and find that the additional contribution of the octupolar electric tidal parameter and quadrupolar magnetic tidal parameter or the change of multipole Love relation has no significant impact on the inferred NS properties. However, adding the $f$-mode dynamical phase lowers the 90% upper bound on $\tilde{\mathrm{\Lambda }}$ by 16–20% as well as lowers the upper bound of NSs radii by $\sim 500\mathrm{m}$. The combined URs (multipole Love and $f$-Love) developed in this work predict a higher median (also a higher 90% upper bound) for $\tilde{\mathrm{\Lambda }}$ by 6% and also predict higher radii for the binary components of GW170817 by 200–300 m compared to the URs used previously in the literature. We further perform injection and recovery studies on simulated events with different EoSs in $\mathrm{A}+$ detector configuration as well as with the third generation Einstein Telescope. In agreement with the literature, we find that neglecting $f$-mode dynamical tides can significantly bias the inferred NS properties, especially for low mass NSs. However, we also find that the impact of the URs is within statistical errors.

Figure 1

(a) EoSs used in this work. (b) Mass radius relation corresponding to the EoSs used in this work. For each EoS, the corresponding M-R relationship is shown up to the maximum possible stable mass. Horizontal bands correspond to masses $M={2.072}_{-0.066}^{+0.067}{M}_{\odot }$ of PSR $\mathrm{J}0740+6620$ [70] and $M={2.01}_{-0.04}^{+0.04}{M}_{\odot }$ of PSR $\mathrm{J}0348+0432$ [71]. The mass radius estimates of the two companion neutron stars in the merger event GW170817 [20] are shown by the shaded area labeled with GW170817 M1 (M2) [72].

Figure 2

Showing universal relations of higher order tidal deformabilities with ${\mathrm{\Lambda }}_2$ along with the relative errors corresponding to fit relation (a) for ${\mathrm{\Lambda }}_3$ and (b) for ${\mathrm{\Lambda }}_4$.

Figure 3

Showing universal relations of tidal deformabilities of magnetic type with ${\mathrm{\Lambda }}_2$ along with the relative errors corresponding to fit relation (a) for ${\mathrm{\Sigma }}_2$ and (b) for ${\mathrm{\Sigma }}_3$.

Figure 4

Top: universal relations among stellar compactness ($C$) and dimensionless quadrupolar tidal deformability (${\mathrm{\Lambda }}_2$). Bottom: relative error corresponding to fit relation.

Figure 5

Top: variation of mass scaled angular frequency of $f$-mode as a function of ${\mathrm{\Lambda }}_2$ along with the fit relation. Bottom: relative error corresponding to fit relation (a) for quadrupolar ($\ell =2$) $f$-mode frequency and ${\mathrm{\Lambda }}_2$, (b) for $M{\omega }_3-{\mathrm{\Lambda }}_2$, and (c) for $M{\omega }_4-{\mathrm{\Lambda }}_2$ UR.

Figure 6

The one-dimensional marginalized posterior distributions of $\tilde{\mathrm{\Lambda }}$, recovered with the different model corrections and different URs.

Figure 7

(a) The radius of the primary component ($m_1$, heavier one) of the binary, estimated through $C-{\mathrm{\Lambda }}_2$ URs from the tidal parameter (${\mathrm{\Lambda }}_{2,1}$) and mass distribution ($m_1$). (b) Same as (a) but for the lighter component ($m_2$). Different URs used are labeled in the superscript to the name of waveform models.

Figure 8

Histogram and posterior distribution of recovered $\tilde{\mathrm{\Lambda }}$ for different injection events with $\mathrm{A}+$ detector configuration corresponding to the (a) Soft-EoS, (b) APR4 EoS, and (c) Stiff-EoS. The injected values are shown with black dashed lines.

Figure 9

Relative errors for higher order electric tidal parameters (${\mathrm{\Lambda }}_3$, ${\mathrm{\Lambda }}_4$), magnetic tidal parameter ${\mathrm{\Sigma }}_2$, stellar compactness ($C$), and mass scaled $f$-mode angular frequency ($M\omega$) resulting from different URs in the range ${\mathrm{\Lambda }}_2\le {10}^4$. Solid (dashed) line represents the upper limit on the 99% (90%) of the errors. Different colored line represents different URs as labeled in the figures. If necessary, for better visibility the $y$-axis is scaled with logarithmic scale.

Figure 10

Histogram and posterior distribution of recovered $\tilde{\mathrm{\Lambda }}$ for different injection events corresponding to the Soft-EoS (upper panel), APR4 EoS (middle panel), and Stiff-EoS (lower panel). The injected values are shown with black dashed lines. The analysis is done for ET configuration.