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In this paper, we aim at using the DECi-hertz InterferometerGravitational-wave Observatory (DECIGO), a future Japanese spacegravitational-wave antenna sensitive to frequency range between LISA andground-based detectors, to provide gravitational-wave constraints on the cosmiccurvature at $z\sim 5$. In the framework of the well-known distance sum rule,the perfect redshift coverage of the standard sirens observed by DECIGO,compared with lensing observations including the source and lens from LSST,makes such cosmological-model-independent test more natural and general.Focusing on three kinds of spherically symmetric mass distributions for thelensing galaxies, we find that the cosmic curvature is expected to beconstrained with the precision of $\Delta \Omega_K \sim 10^{-2}$ in the earlyuniverse ($z\sim5.0$), improving the sensitivity of ET constraints by about afactor of 10. However, in order to investigate this further, the mass densityprofiles of early-type galaxies should be properly taken into account.Specially, our analysis demonstrates the strong degeneracy between the spatialcurvature and the lens parameters, especially the redshift evolution ofpower-law lens index parameter. When the extended power law mass densityprofile is assumed, the weakest constraint on the cosmic curvature can beobtained. Whereas, the addition of DECIGO to the combination of LSST+DECIGOdoes improve the constraint on the luminosity density slope and the anisotropyof the stellar velocity dispersion significantly. Therefore, our paperhighlights the benefits of synergies between DECIGO and LSST in constrainingnew physics beyond the standard model, which could manifest itself throughaccurate determination of the cosmic curvature.

DECi-hertz Interferometer Gravitational-wave Observatory: Forecast constraints on the cosmic curvature with LSST strong lenses