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 non-resonant 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 $p$-$g$ model to a standard one. We find that the observed signal is consistent with waveform models that neglect $p$-$g$ 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 $p$-$g$ effects and recovering them with the $p$-$g$ model, we show that there is a $\simeq 50\mathrm{\%}$ probability of obtaining similar $\ln B^{pg}{!pg}$evenwhen$p$-$g$effectsareabsent\mathrm{.}Wefindthatthe$p$-$g$amplitudefor1.4$M\odot$neutronstarsisconstrainedto$\lesssim $\text{few}$\times10^{-7}$,withmaximaaposteriorinear$\sim 10^{-7}$and$p$-$g$saturationfrequency$\sim 70, $\mathrm{H}\mathrm{z}$$. This suggests that there are less than a few hundred excited modes, assuming they all saturate by wave breaking. For comparison, theoretical upper bounds suggest a $p$-$g$ amplitude $\lesssim {10}^{-6}$ and $\lesssim {10}^3$ modes saturating by wave breaking. Thus, the measured constraints only rule out extreme values of the $p$-$g$ parameters. They also imply that the instability dissipates $\lesssim 10^{51}, $\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{s}$$ over the entire inspiral, i.e., less than a few percent of the energy radiated as gravitational waves.