New dynamical tide constraints from current and future gravitational wave detections of inspiralling neutron stars

Previous theoretical works using the premerger orbital evolution of coalescing neutron stars to constrain properties of dense nuclear matter assume a gravitational wave phase uncertainty of a few radians, or about a half cycle. However, recent studies of the signal from GW170817 and next-generation detector sensitivities indicate actual phase uncertainties at least twenty times better. Using these refined estimates, we show that future observations of nearby sources like GW170817 may be able to reveal neutron star properties beyond just radius and tidal deformability, such as the matter composition and/or presence of a superfluid inside neutron stars, via tidal excitation of g-mode oscillations. Data from GW170817 already limits the amount of orbital energy that is transferred to the neutron star to $<2\times {10}^{47}\mathrm{erg}$ and the g-mode tidal coupling to $Q_{\alpha }<{10}^{-3}$ at 50 Hz ($5\times {10}^{48}\mathrm{erg}$ and $4\times {10}^{-3}$ at 200 Hz), and future observations and detectors will greatly improve upon these constraints. In addition, analysis using general parametrization models that have been applied to the so-called p-g instability show that the instability already appears to be restricted to regimes where the mechanism is likely to be inconsequential; in particular, we show that the number of unstable modes is $\ll 100$ at $\lesssim 100\mathrm{Hz}$, and next generation detectors will essentially rule out this mechanism (assuming that the instability remains undetected). Finally, we illustrate that measurements of tidal excitation of r-mode oscillations in nearby rapidly rotating neutron stars are within reach of current detectors and note that even nondetections will limit the inferred inspiralling neutron star spin rate to $<20\mathrm{Hz}$, which will be useful when determining other parameters such as neutron star mass and tidal deformability.

Figure 1

Phase uncertainty as a function of GW frequency $f$. Shaded regions indicate $\pm 1\sigma$ deviation of $\mathrm{\Delta }\mathrm{\Phi }$ from a GW170817 waveform ([16]; see text) and from a model waveform weighted by the sensitivity of $\mathrm{A}+$ and Cosmic Explorer (CE) [3] at a distance of 40 Mpc. Dotted lines indicate $\pm \mathrm{\Delta }\mathrm{\Phi }$ assuming (dimensionless) overlap integral $Q_{\alpha }={10}^{-3}$ and NS radius $R=12\mathrm{km}$ [see Eq. (7)].

Figure 2

Relative energy transfer $|\mathrm{\Delta }E/E_{\mathrm{orb}}|$ as a function of GW frequency $f$. Thick lines are upper limits calculated using $\pm \mathrm{\Delta }\mathrm{\Phi }$ from Fig. 1 and Eq. (1) for GW170817 (short-dashed), $\mathrm{A}+$ (long-dashed), and Cosmic Explorer (CE; solid). Long-dashed-dotted and short-dashed-dotted lines are for total energy transferred to normal and superfluid g-modes, respectively (see text for details), while the dotted line is for total energy dissipated by the p-g instability assuming NS radius $R=12\mathrm{km}$ and $N_0\beta \lambda =1$ [see Eq. (8)] and the short-long-dashed line is for energy transferred to the orbit by r-modes and $R=12\mathrm{km}$ [see Eq. (12)]. Note that for the r-mode, each particular GW frequency corresponds to a specific NS spin frequency [here we assumed $f=(4/3)f_{\mathrm{spin}}$].

Figure 3

Dimensionless overlap integral $Q_{\alpha }$ as a function of GW frequency $f$. Lines are upper limits calculated using $\pm \mathrm{\Delta }\mathrm{\Phi }$ from Fig. 1 and Eq. (7) with a NS radius $R=12\mathrm{km}$ for GW170817 (short-dashed), $\mathrm{A}+$ (long-dashed), and Cosmic Explorer (CE; solid). Circles are for constant $\mathrm{\Gamma }-\gamma$ stratification g-modes, while the triangle is for a varying stratification g-mode (see text); light dotted lines connect $Q_{\alpha }$ values from g-modes with the same stratification but different radial node $n$. Stars are for g-modes ($n=1$) calculated from NS models motivated by the BSk21 and SLy4 equations of state (see Counsell et al., in prep., and [20], respectively), which have strong internal composition gradients.

Figure 4

Amplitude of p-g instability $A_0$ as a function of GW frequency $f$. Lines are upper limits calculated using $\pm \mathrm{\Delta }\mathrm{\Phi }$ from Fig. 1 and the power law model given by Eq. (9) (left panel) and the asymptotic model given by Eq. (11) (right panel) with frequency power law index $n_0=0$ for GW170817 (short-dashed; light lines are for $n_0=-1,+1,+2$), $\mathrm{A}+$ (long-dashed), and Cosmic Explorer (CE; solid). The vertical line in the right panel indicates the frequency at which mode saturation is assumed to occur, here taken to be $f_0=50\mathrm{Hz}$. Parameters of the p-g instability are the number of unstable modes $N_0$ and mode energy relative to saturation maximum $\beta$ ($\le 1$) and $\lambda$ ($\sim 0.1–1$) is a function of binary separation.