Null test for cosmic curvature using Gaussian process

The cosmic curvature $\Omega_{K,0}$, which determines the spatial geometry of the universe, is an important parameter in modern cosmology. Any deviation from $\Omega_{K,0}=0$ would have a profound impact on primordial inflation paradigm and fundamental physics. In this work, we adopt a cosmological model-independent method to test whether $\Omega_{K,0}$ deviates from zero. We use the Gaussian process to reconstruct the reduced Hubble parameter $E(z)$ and the derivative of distance $D'(z)$ from observational data, and then determine $\Omega_{K,0}$ with a null test relation. The cosmic chronometer (CC) Hubble data, baryon acoustic oscillation (BAO) Hubble data, and supernovae Pantheon sample are considered. Our result is consistent with a spatially flat universe within the domain of reconstruction $0<z<2.3$, at the $1\sigma$ confidence level. In the redshift interval $0<z<1$, the result favors a flat universe, while at $z>1$, it tends to favor a closed universe. In this sense, there is still a possibility for a closed universe. We also carry out the null test of the cosmic curvature at $0<z<4.5$ using the simulated gravitational wave standard sirens, CC+BAO and redshift drift Hubble data. The result shows that in the future, with the synergy of multiple high-quality observations, we can tightly constrain the spatial geometry or exclude the flat universe.