  In this paper, we aim at using the DECi-hertz Interferometer
Gravitational-wave Observatory (DECIGO), a future Japanese space
gravitational-wave antenna sensitive to frequency range between LISA and
ground-based detectors, to provide gravitational-wave constraints on the cosmic
curvature 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 the
lensing galaxies, we find that the cosmic curvature is expected to be
constrained with the precision of $\Delta \Omega_K \sim 10^{-2}$ in the early
universe ($z\sim5.0$), improving the sensitivity of ET constraints by about a
factor of 10. However, in order to investigate this further, the mass density
profiles of early-type galaxies should be properly taken into account.
Specially, our analysis demonstrates the strong degeneracy between the spatial
curvature and the lens parameters, especially the redshift evolution of
power-law lens index parameter. When the extended power law mass density
profile is assumed, the weakest constraint on the cosmic curvature can be
obtained. Whereas, the addition of DECIGO to the combination of LSST+DECIGO
does improve the constraint on the luminosity density slope and the anisotropy
of the stellar velocity dispersion significantly. Therefore, our paper
highlights the benefits of synergies between DECIGO and LSST in constraining
new physics beyond the standard model, which could manifest itself through
accurate determination of the cosmic curvature.
