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Although the spatial curvature has been precisely determined via the cosmicmicrowave background (CMB) observation by Planck satellite, it still suffersfrom the well-known cosmic curvature tension. As a standard siren,gravitational waves (GWs) from binary neutron star mergers provide a direct wayto measure the luminosity distance. In addition, the accelerating expansion ofthe universe may cause an additional phase shift in the gravitational waveform,which allows us to measure the acceleration parameter. This measurementprovides an important opportunity to determine the curvature parameter$\Omega_k$ in the GW domain based on the combination of two differentobservables for the same objects at high redshifts. In this study, weinvestigate how such an idea could be implemented with future generation ofspace-based DECi-hertz Interferometer Gravitational-wave Observatory (DECIGO)in the framework of two model-independent methods. Our results show that DECIGOcould provide a reliable and stringent constraint on the cosmic curvature at aprecision of $\Delta\Omega_k$=0.12, which is comparable to existing resultsbased on different electromagnetic data. Our constraints are more stringentthan the traditional electromagnetic method from the Pantheon SNe Ia sample,which shows no evidence for the deviation from the flat universe at $z\sim2.3$. More importantly, with our model-independent method, such asecond-generation space-based GW detector would also be able to explore thepossible evolution $\Omega_k$ with redshifts, through direct measurements ofcosmic curvature at different redshifts ($z\sim 5$). Such a model-independent$\Omega_k$ reconstruction to the distance past can become a milestone ingravitational-wave cosmology.

Multiple measurements of gravitational waves acting as standard probes: model-independent constraints on the cosmic curvature with DECIGO