Constraining the $p$-Mode–$g$-Mode Tidal Instability with GW170817

We analyze the impact of a proposed tidal instability coupling $p$ modes and $g$ modes within neutron stars on GW170817. This nonresonant instability transfers energy from the orbit of the binary to internal modes of the stars, accelerating the gravitational-wave driven inspiral. We model the impact of this instability on the phasing of the gravitational wave signal using three parameters per star: an overall amplitude, a saturation frequency, and a spectral index. Incorporating these additional parameters, we compute the Bayes factor ($\ln B_{!pg}^{pg}$) comparing our $p\text{−}g$ model to a standard one. We find that the observed signal is consistent with waveform models that neglect $p\text{−}g$ effects, with $\ln B_{!pg}^{pg}=0.03_{-0.58}^{+0.70}$ (maximum a posteriori and 90% credible region). By injecting simulated signals that do not include $p\text{−}g$ effects and recovering them with the $p\text{−}g$ model, we show that there is a $\simeq 50\%$ probability of obtaining similar $\ln B_{!pg}^{pg}$ even when $p\text{−}g$ effects are absent. We find that the $p\text{−}g$ amplitude for $1.4M_{\odot }$ neutron stars is constrained to less than a few tenths of the theoretical maximum, with maxima a posteriori near one-tenth this maximum and $p\text{−}g$ saturation frequency $\sim 70\mathrm{Hz}$. This suggests that there are less than a few hundred excited modes, assuming they all saturate by wave breaking. For comparison, theoretical upper bounds suggest $\lesssim {10}^3$ modes saturate by wave breaking. Thus, the measured constraints only rule out extreme values of the $p\text{−}g$ parameters. They also imply that the instability dissipates $\lesssim 10^{51}\mathrm{erg}$ over the entire inspiral, i.e., less than a few percent of the energy radiated as gravitational waves.

Figure 1

Distributions of $\ln B_{!pg}^{pg}$ due to sampling uncertainty when analyzing GW170817 data with different values of $f_{\mathrm{low}}$. The solid red curves assume high-spin priors ($|{\chi }_i|\le 0.89$) and the dashed blue curves assume low-spin priors ($|{\chi }_i|\le 0.05$).

Figure 2

Posterior distributions for ${\mathcal{H}}_{!pg}$ (red) and ${\mathcal{H}}_{pg}$ with Hanford, Livingston, and Virgo data (thick black, gray shading), Hanford data only (dark blue), and Livingston data only (light blue) using GW data above 30 Hz, $|{\chi }_i|\le 0.89$, and a uniform-in-${\log }_{10}{A}_0$ prior. Left: a subset of parameters shared by ${\mathcal{H}}_{!pg}$ and ${\mathcal{H}}_{pg}$. Right: a subset of parameters belonging only to ${\mathcal{H}}_{pg}$. We only show one-dimensional posteriors for the single instrument data, although the multidimensional posteriors are similarly consistent with the full ${\mathcal{H}}_{pg}$ data. Contours in the two-dimensional distributions represent 10%, 50%, and 90% confidence regions.

Figure 3

Upper limits on the cumulative energy dissipated by the $p\text{−}g$ instability as a function of frequency. We note the relatively strong constraints at lower frequencies where $p\text{−}g$ effects are more pronounced.