Multiband Gravitational Wave Detection Prospects for M31 UCXB-1 System in Low and Middle Frequency Band

The recent discovery of M31 UCXB-1, the first extragalactic ultracompact X-ray binary (UCXB) with an orbital period of $T_{\rm orb} \sim 465$ s, presents a unique laboratory for studying close binary evolution and an unprecedented target for continuous gravitational wave (GW) searches. Its identification as a strong candidate black hole-white dwarf (BH-WD) system, combined with its exceptionally short period and high X-ray luminosity, suggests it may be one of the most vital low-frequency GW sources in M31. In this \textit{Letter}, we investigate the detectability of its GW signal for future space-borne detectors in multiband GW detection. We find that while its signal-to-noise ratio (S/N) for low-frequency detectors remains marginal for high-confidence detection, middle-frequency detectors such as DECIGO and BBO are far more promising, potentially achieving S/N $\varrho>8$ within reasonable observational duration. With a primary mass of only $m_1 > 5.4M_\odot$ (or $6.6M_\odot$), the network of all low and middle frequency detector (or BBO alone) is capable of detecting GW from this system with a $\varrho > 8$, during 10-year observation. Furthermore, orbital eccentricity can enhance the GW strain at higher harmonics, further improving its detectability, especially for middle-frequency detectors. This study establishes M31 UCXB-1 as a key prototype of short-period UCXBs, cementing its role as a cornerstone for multiband, multi-messenger astrophysics and a vital bridge between X-ray astronomy and the future GW era.