General Cosmography Model with Spatial Curvature

The cosmographic approach is adopted to determine the spatial curvature (i.e., $Ω_K$) combining the latest released cosmic chronometers data (CC), the Pantheon sample of type Ia supernovae observations, and the baryon acoustic oscillation measurements. We use the expanded transverse comoving distance $D_M(z)$ as a basic function for deriving $H(z)$ and the other cosmic distances. In this scenario, $Ω_K$ can be constrained only by CC data. To overcome the convergence issues at high-redshift domains, two methods are applied: the Padé approximants and the Taylor series in terms of the new redshift $y=z/(1+z)$. Adopting the Bayesian evidence, we find that there is positive evidence for the Padé approximant up to order ($2,2$) and weak evidence for the Taylor series up to 3-rd order against $Λ\text{CDM}+Ω_K$ model. The constraint results show that a closed universe is preferred by the present observations under all the approximants used in this study. And the tension level of the Hubble constant $H_0$ is less than $2σ$ significance between different approximants and the local distance ladder determination. For each assumed approximant, $H_0$ is anti-correlated with $Ω_K$ and the sound horizon at the end of the radiation drag epoch, which indicates that the $H_0$ tension problem can be slightly relaxed by introducing $Ω_K$ or any new physics which can reduce the sound horizon in the early universe.

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