A dark energy parameterization independent constraint of the spatial curvature $\Omega_K$

Determining the spatial curvature $\Omega_K$ of the Universe has long been crucial in cosmology. In practice, this effort is often entangled with assumptions of dark energy. A combination of distance ($D_{\rm M}$, $D_{\rm L}$) and expansion rate ($H(z)$) measurements can break this degeneracy. However, fitting against discrete data points requires parameterizations of distance and expansion rate as functions of redshifts, which often induces cosmological model dependence. In this work, we propose a new dark energy model-independent parameterization of the cosmological comoving radial distance $\chi$. Fitting data combining distance ($D_{\rm M}$, $D_{\rm L}$) and Hubble parameter (or equivalently $D_H$) measurements, we are then able to obtain $\Omega_K$ in a dark energy model-independent manner. We test this parameterization and the associated fitting scheme with mock data generated with a wide range of fiducial dark energy equations of state ($-1.3<w<1.3$), finding that the best-fit $\Omega_K$ is always unbiased. Then we combine SDSS Baryon Acoustic Oscillation (BAO), Pantheon+ sample of Type Ia Supernovae (SNe Ia), and Observational Hubble Data (OHD) to constrain $\Omega_K$. We find a flat universe with $\Omega_K=-0.01\pm 0.09$. Most constraining power is contributed by SDSS BAO, with the BAO-alone constraint $\Omega_K=-0.03 \pm 0.10$. When replacing SDSS BAO with DESI year-one BAO measurement, we obtain $\Omega_K=0.06 \pm 0.08$. With the full DESI BAO data alone, we forecast $\sigma(\Omega_K)\sim 0.03$. Our result verifies the flatness of the universe free of dark energy modeling, and the proposed parameterization would be useful for future investigation of $\Omega_K$ and other parameters of interest, such as the horizon radius.