Using gravitational-wave data to constrain dynamical tides in neutron star binaries

We discuss the role of dynamical tidal effects for inspiraling neutron star binaries, focusing on features that may be considered “unmodeled” in gravitational-wave searches. In order to cover the range of possibilities, we consider (i) individual oscillation modes becoming resonant with the tide, (ii) the elliptical instability, where a pair of inertial modes exhibit a nonlinear resonance with the tide, and (iii) the nonresonant p-g instability which may arise as high-order pressure (p) and gravity (g) modes in the star couple nonlinearly to the tide. In each case, we estimate the amount of additional energy loss that needs to be associated with the dynamical tide in order for the effect to impact on an observed gravitational-wave signal. We explore to what extent the involved neutron star physics may be considered known and how one may be able to use observational data to constrain theory.

Figure 1

A schematic illustration of the impact of tidal deformability on a binary neutron star signal. We show the estimated shift in the number of gravitational wave cycles $|\mathrm{\Delta }\mathcal{N}|$ as a function of the gravitational-wave frequency $f$. The grey band follows from (17) if we assume a Newtonian $n=1$ polytrope (for which $k_2\approx 0.26$), two equal $1.4M_{\odot }$ neutron stars (thus doubling the value of $\mathrm{\Delta }\mathcal{N}$), and the “reasonable” range of radii 10–14 km. The dashed curves show how this band shifts if we consider the (likely unrealistic) case of two $1.1M_{\odot }$ stars. (For more detailed/realistic models, see, for example, Fig. 4 in [18].) The dashed horizontal line represents the indicative level of $|\mathrm{\Delta }\mathcal{N}|\approx 0.5$ above which the effect will leave an imprint in a matched filter search, and the vertical shaded region represents our example frequency range between 30 and 300 Hz.

Figure 2

Constraints on $Q_{\alpha }$ if a limit $|\mathrm{\Delta }\mathcal{N}|\le 0.5$ were to be inferred from inspiral data. The thin black lines represent equal mass $1.4M_{\odot }$ binaries with neutron star radius 10 km (upper curve) and 14 km (lower curve). The grey region represents the expected radius range from x-ray observations [43]. As an indication, the thick horizontal (red) line represents the largest values of $Q_{\alpha }$ for the g modes of a nonrotating star from [29]. This should be taken as indicative of what is expected from theory (with the caveats discussed in the main text). Finally, the shaded vertical region relates to an example where the observational constraint is obtained for a distinct frequency band (here taken to be 100–150 Hz). This figure illustrates that the resonant modes of a nonrotating star may be difficult to detect, but there could be a relevant effect below 50 Hz or so, if the neutron star radius were to be surprisingly large (the dashed curve shows the result for a radius of 17 km). One should also keep in mind that rotation may lead to slightly larger values of $Q_{\alpha }$, in which case the chance of detection would improve.

Figure 3

Comparing the different frequency constraints to see if the elliptical instability may play a role for neutron star binaries. The vertical axis is the spin frequency, $f_s$, while the horizontal one is the gravitational-wave frequency, $f$. The lower horizontal dashed line represents PSR J0737-3039A (extrapolated to be at 35 Hz as it enters the LIGO sensitivity band). A system has to be above all the other curves, arising from (46), (47), and (55) (showing both $T={10}^6\mathrm{K}$ and $10^7\mathrm{K}$), in order to be unstable. These estimates suggest that a system like PSR J0737-3039A may (briefly) exhibit the elliptical instability during the inspiral phase. Any (at the moment fiducial) faster spinning pulsar could be more significantly affected.

Figure 4

A schematic illustration of the impact the p-g instability may have on a binary neutron star signal. Left panel: The estimated shift in the accumulated number of gravitational wave cycles $|\mathrm{\Delta }\mathcal{N}|$ as a function of the gravitational-wave frequency $f$. The grey band recalls the result for the tidal compressibility from Fig. 1. The dashed horizontal line represents the indicative level of $|\mathrm{\Delta }\mathcal{N}|\approx 0.5$ above which the effect would leave an imprint in a matched filter search. The solid curves show the estimate effect of the p-g instability [taking $f=f_b$ in (75)]. The curves represent (taking the unknown parameters $\lambda =\beta =1$ for simplicity), cases where $N=1$, 10, and 25 modes (as indicated) are excited by the instability. Given the uncertainties in the model, these estimates should be taken with a fair bit of caution. Finally, the shaded region at low frequencies represents the estimated level of resonant r modes, given by (35). Right panel: The same, but for $|d\mathrm{\Delta }\mathcal{N}/df|$. This provides a clearer idea of how one can distinguish between the two mechanisms by comparing the effect in a sequence of frequency ranges (time windows). A more realistic model of the p-g instability would account for the finite growth time, which would smooth out the curves, but we would still expect them to be strongly peaked.