Effects of orbital eccentricity on continuous gravitational waveforms from triaxially deformed precessing neutron stars in tight binaries

The successful detection of continuous gravitational waves from spinning neutron stars (NSs) will shape our understanding of the physical properties of dense matter under extreme conditions. Binary population synthesis simulations show that forthcoming space-borne gravitational wave detectors may be capable of detecting some tight Galactic double NSs with 10-min orbital periods. Successfully searching for continuous waves from the individual NS in such a close binary demands extremely precise waveform templates considering the interaction between the NS and its companion. Unlike the isolated formation channel, double NS systems from the dynamical formation channel have moderate to high orbital eccentricities. To accommodate these systems, we generalize the analytical waveforms from triaxial nonaligned NSs under spin-orbit coupling derived by Feng \textit{et al.} [\href{https://journals.aps.org/prd/abstract/10.1103/PhysRevD.108.063035}{Phys. Rev. D 108, 063035 (2023)}] to incorporate the effects of the orbital eccentricity. Our findings suggest that for binaries formed through isolated binary evolution, the impact of eccentricity on the continuous waves of their NSs can be neglected. In contrast, for those formed through dynamical processes, it is necessary to consider eccentricity, as high-eccentricity orbits can result in a fitting factor of $\lesssim 0.97$ (0.9) within approximately 0.5 (1) to 2 (5) yr of a coherent search (at wave frequencies of 100 and 200 Hz). Once the continuous waves from spinning NSs in tight binaries are detected, the relative measurement accuracy of eccentricity can reach $\Delta e / e \sim O(10^{-7})$ for a signal-to-noise ratio of $O(100)$ based on the Fisher information matrix, bearing significant implications for understanding the formation mechanisms of double NS systems.