Using the fσ8(z) redshift space distortion (RSD) data, the $\sigma _8^0\!-\!\Omega _\mathrm{ m}^0$ tension is studied utilizing a parametrization of growth rate f(z) = Ωm(z)γ. Here, f(z) is derived from the expansion history H(z) which is reconstructed from the observational Hubble data applying the Gaussian process method. It is found that different priors of H0 have great influences on the evolution curve of H(z) and the constraint of $\sigma _8^0\!-\!\Omega _\mathrm{ m}^0$. When using a larger H0 prior, the low redshifts H(z) deviate significantly from that of the ΛCDM model, which indicates that a dark energy model different from the cosmological constant can help to relax the H0 tension problem. The tension between our best-fitting values of $\sigma _8^0\!-\!\Omega _\mathrm{ m}^0$ and that of the Planck 2018 ΛCDM (PLA) will disappear (less than 1σ) when taking a prior for H0 obtained from PLA. Moreover, the tension exceeds 2σ level when applying the prior H0 = 73.52 ± 1.62 km s−1 Mpc−1 resulted from the Hubble Space Telescope photometry. By comparing the $S_8\!-\!\Omega _\mathrm{ m}^0$ planes of our method with the results from KV450+DES-Y1, we find that using our method and applying the RSD data may be helpful to break the parameter degeneracies.

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 ΛCDM + 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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