Eccentricity-tide coupling: impact on binary neutron stars and extreme mass-ratio inspirals

We study the effect of tidal interaction between two compact bodies in an eccentric orbit. We assume the tidal fields to be static. Therefore, we ignore the dynamic tides and resonant excitations. Using the results, we find the analytical expression for the phase shift of the emitted gravitational wave. In the process, we find that in the leading order, the initial eccentricity $e_0$ and the dimensionless tidal deformability $\Lambda$ couple as $\sim e_0^n\Lambda$, where $n$ is a positive number. We only focus on the dominant contribution, i.e., $e_0^2\Lambda$. We also compute the accumulated dephasing for binary neutron star systems. We find that for optimistic values of eccentricities $e_0 \sim .05$ and $\Lambda \sim 600$, the accumulated dephasing is $\mathcal{O}(10^{-4})$ radian, requiring a signal-to-noise ratio $\sim 7000$ to be observable. Therefore, these effects can be measured in binary neutron star systems with large eccentricities if the signal-to-noise ratios of the systems are also very large. Hence, in third-generation detectors, it may have an observable impact if the systems have large eccentricities. We also explore the impact of this effect on extreme mass-ratio inspirals (EMRIs). We find that even for supermassive bodies with small values of $\Lambda \sim 10^{-3}$, this effect has large dephasing in EMRIs $\sim \mathcal{O}(10)$ radian. Therefore, this effect will help in probing the nature of the supermassive bodies in an EMRI.