Constraining the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>p</mml:mi></mml:math>-Mode–<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>g</mml:mi></mml:math>-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 ($\mathrm{ln}{B}_{!pg}^{pg}$) comparing our $p\text{\ensuremath{-}}g$ model to a standard one. We find that the observed signal is consistent with waveform models that neglect $p\text{\ensuremath{-}}g$ effects, with $\mathrm{ln}{B}_{!pg}^{pg}=0.0{3}_{\ensuremath{-}0.58}^{+0.70}$ (maximum a posteriori and 90% credible region). By injecting simulated signals that do not include $p\text{\ensuremath{-}}g$ effects and recovering them with the $p\text{\ensuremath{-}}g$ model, we show that there is a $\ensuremath{\simeq}50%$ probability of obtaining similar $\mathrm{ln}{B}_{!pg}^{pg}$ even when $p\text{\ensuremath{-}}g$ effects are absent. We find that the $p\text{\ensuremath{-}}g$ amplitude for $1.4\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ neutron stars is constrained to less than a few tenths of the theoretical maximum, with maxima a posteriori near one-tenth this maximum and $p\text{\ensuremath{-}}g$ saturation frequency $\ensuremath{\sim}70\text{ }\text{ }\mathrm{Hz}$. 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 $\ensuremath{\lesssim}{10}^{3}$ modes saturate by wave breaking. Thus, the measured constraints only rule out extreme values of the $p\text{\ensuremath{-}}g$ parameters. They also imply that the instability dissipates $\ensuremath{\lesssim}1{0}^{51}\text{ }\text{ }\mathrm{erg}$ over the entire inspiral, i.e., less than a few percent of the energy radiated as gravitational waves.

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