Tidal deformability of neutron stars with realistic equations of state and their gravitational wave signatures in binary inspiral

TL;DR

This work assesses how the tidal deformability $\lambda$ of neutron stars, encapsulated by the Love number $k_2$ and radius $R$ via $\lambda = \frac{2}{3G} k_2 R^5$, imprints the gravitational-wave phase in the early inspiral and can constrain the nuclear EOS. It computes $k_2$ and $\lambda$ for a broad set of realistic EOS by solving the relativistic perturbation of the Oppenheimer–Volkoff structure, including a surface-density correction for self-bound stars, and finds that while $k_2(m)$ varies modestly, $\lambda(m)$ spans an order of magnitude across EOS. Using a PN framework plus leading tidal corrections and Fisher-matrix estimates, the study shows that tidal effects are detectable in third-generation detectors like the Einstein Telescope but are generally inaccessible to Advanced LIGO at 100 Mpc except for unusually stiff EOS or nearby events. The results imply that future GW observations can significantly constrain the NS EOS by measuring $\tilde{\lambda}$, with the potential to rule out large regions of EOS parameter space as detector sensitivities improve.

Abstract

The early part of the gravitational wave signal of binary neutron star inspirals can potentially yield robust information on the nuclear equation of state. The influence of a star's internal structure on the waveform is characterized by a single parameter: the tidal deformability lambda, which measures the star's quadrupole deformation in response to the companion's perturbing tidal field. We calculate lambda for a wide range of equations of state and find that the value of lambda spans an order of magnitude for the range of equation of state models considered. An analysis of the feasibility of discriminating between neutron star equations of state with gravitational wave observations of the early part of the inspiral reveals that the measurement error in lambda increases steeply with the total mass of the binary. Comparing the errors with the expected range of lambda, we find that Advanced LIGO observations of binaries at a distance of 100 Mpc will probe only unusually stiff equations of state, while the proposed Einstein Telescope is likely to see a clean tidal signature.

Figures (4)

• Figure 1: Top panel: Love number as a function of compactness. Gray dotted curves are energy density polytropes ($p=K\epsilon^{1+1/n}$), and gray solid curves are rest-mass density polytropes ($p=K\rho^{1+1/n}$). Both polytropes are the same for $n=0$. EOS with only $npe\mu$ matter are solid and those that also incorporate $\pi$/hyperon/quark matter are dot-dashed. The three SQM EOS are dashed and overlap. They approach the $n=0$ curve at low compactness, where $k_2$ has a maximum value of 0.75 as $m/R \to 0$. Bottom panel: Love number as a function of mass for the same set of realistic EOS. Note that there is more variation in $k_2$ between different EOS for fixed mass than for fixed compactness.
• Figure 2: Tidal deformability $\lambda$ of a single neutron star as a function of neutron-star mass for a range of realistic EOS. The top figure shows EOS that only include $npe\mu$ matter; the middle figure shows EOS that also incorporate $\pi$/hyperon/quark matter; the bottom figure shows strange quark matter EOS. The dashed lines between the various shaded regions represent the expected uncertainties in measuring $\lambda$ for an equal-mass binary inspiral at a distance of $D = 100$ Mpc as it passes through the gravitational wave frequency range 10 Hz--450 Hz. Observations with Advanced LIGO will be sensitive to $\lambda$ in the unshaded region, while the Einstein Telescope will be able to measure $\lambda$ in the unshaded and light shaded regions. See text below.
• Figure 3: Weighted $\tilde{\lambda}$ for a range of chirp mass $\mathcal{M}$ and dimensionless reduced mass $\eta$, for three of the EOSs considered above. The values of $\eta$ equal to {0.25, 0.242, 0.222} correspond to the mass ratios $m_2/m_1 =$ {1.0, 0.7, 0.5}. Also plotted (as in Fig. \ref{['fig:lambdaofm']}) are the uncertainties $\Delta\tilde{\lambda}$ in measuring $\tilde{\lambda}$ for a binary at 100 Mpc between 10 Hz--450 Hz. The solid, dashed, and dotted curves correspond to $\Delta\tilde{\lambda}$ for $\eta= 0.25$, 0.242, and 0.222 respectively.
• Figure 4: The reduction in accumulated gravitational wave phase due to tidal effects, $\Phi_{3.5,PP}(f_{GW}) - \Phi_{3.5,\lambda}(f_{GW})$, is plotted with thick lines as a function of gravitational wave frequency, for a range of $\lambda$ appropriate for realistic neutron star EOS and the masses considered. The 3.5 post-Newtonian TaylorT4 PN specification is used as the point-particle reference for the phase calculations. For reference, the difference in accumulated phase between 3.0 and 3.5 post-Newtonian orders of T4 (thin dashed line), and the difference between 3.5 post-Newtonian T4 and 3.5 post-Newtonian T1 (thin dotted line) are also shown. Phase accumulations are integrated from a starting frequency of 10 Hz.