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Register a new userGrowth rate of the tidal p-mode g-mode instability in coalescing binary neutron stars
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Nevin N. Weinberg
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We recently described an instability due to the nonlinear coupling of p-modes to g-modes and, as an application, we studied the stability of the tide in coalescing binary neutron stars. Although we found that the tide is p-g unstable early in the inspiral and rapidly drives modes to large energies, our analysis only accounted for three-mode interactions. Venumadhav, Zimmerman, and Hirata showed that four-mode interactions must also be accounted for as they enter into the analysis at the same order. They found a near-exact cancellation between three- and four-mode interactions and concluded that while the tide in binary neutron stars can be p-g unstable, the growth rates are not fast enough to impact the gravitational wave signal. Their analysis assumes that the linear tide is incompressible, which is true of the static linear tide (the m=0 harmonic) but not the non-static linear tide (m=+/- 2). Here we account for the compressibility of the non-static linear tide and find that the three- and four-mode interactions no longer cancel. As a result, we find that the instability can rapidly drive modes to significant energies (there is time for several dozen e-foldings of growth before the binary merges). We also show that linear damping interferes with the cancellation and may further enhance the p-g growth rates. The early onset of the instability (at gravitational wave frequencies near 50 Hz), the rapid growth rates, and the large number of unstable modes (> 10^3), suggest that the instability could impact the phase evolution of gravitational waves from binary neutron stars. Assessing its impact will require an understanding of how the instability saturates and is left to future work.
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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 $ln B^{pg}_{!pg} = 0.03^{+0.70}_{-0.58}$ (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 $simeq 50%$ probability of obtaining similar $ln B^{pg}_{!pg}$ even when $p$-$g$ effects are absent. We find that the $p$-$g$ amplitude for 1.4 $M_odot$ neutron stars is constrained to $lesssim text{few}times10^{-7}$, with maxima a posteriori near $sim 10^{-7}$ and $p$-$g$ 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 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 $lesssim 10^{51}, mathrm{ergs}$ over the entire inspiral, i.e., less than a few percent of the energy radiated as gravitational waves.
A weakly nonlinear fluid wave propagating within a star can be unstable to three-wave interactions. The resonant parametric instability is a well-known form of three-wave interaction in which a primary wave of frequency $omega_a$ excites a pair of secondary waves of frequency $omega_b+omega_c simeq omega_a$. Here we consider a nonresonant form of three-wave interaction in which a low-frequency primary wave excites a high-frequency p-mode and a low-frequency g-mode such that $omega_b+omega_c >>omega_a$. We show that a p-mode can couple so strongly to a g-mode of similar radial wavelength that this type of interaction is unstable even if the primary wave amplitude is small. As an application, we analyze the stability of the tide in coalescing neutron star binaries to p-g mode coupling. We find that the equilibrium tide and dynamical tide are both p-g unstable at gravitational wave frequencies f_{gw} > 20 Hz and drive p-g pairs to significant energies on very short timescales (much less than the orbital decay time). Resonant parametric coupling to the tide is, by contrast, either stable or drives modes at a much smaller rate. We do not solve for the saturation of the instability and therefore cannot say precisely how it influences neutron star binaries. However, we show that if even a single daughter mode saturates near its wave breaking amplitude, the p-g instability of the equilibrium tide: (i) induces significant orbital phase errors ($Delta phi$ > 1 radian) that accumulate primarily at low frequencies (f_{gw} < 50 Hz) and (ii) heats the neutron star core to T~10^{10} K. Since there are >100 unstable daughters, $Delta phi$ and T are potentially much larger than these values. Tides might therefore significantly influence the gravitational wave signal and electromagnetic emission from neutron star binaries at much larger orbital separations than previously thought.
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