Testing the consistency of early and late cosmological parameters with BAO and CMB data

The recent local measurements of the Hubble constant $H_0$, indicate a significant discrepancy of over 5$\sigma$ compared to the value inferred from \textit{Planck} observations of the cosmic microwave background (CMB). In this paper, we try to understand the origin of this tension by testing the consistency of early and late cosmological parameters in the same observed data. In practice, we simultaneously derive the early and late parameters using baryon acoustic oscillation (BAO) measurements, which provide both low and high-redshift information. To resolve parameter degeneracy, the complementary data from CMB observations are included in the analysis. By using the parameter $\omega_m = \Omega_mh^2$, we introduce ${\rm ratio}(\omega_m)$, defined as the ratio of $\omega_m$ which are constrained from high and low-redshift measurements respectively, to quantify the consistency between early and late parameters. We obtained a value of ${\rm ratio}(\omega_m) = 1.0069\pm0.0070$, indicating there is no tension between early parameters and late parameters in the framework of $\Lambda$CDM model. As a result, the Hubble tension may arise from the differences of datasets or unknown systematic errors in the current data. In addition, we forecast the future BAO measurements of ${\rm ratio}(\omega_m)$, using several galaxy redshift surveys and 21 cm intensity mapping surveys, and find that these measurements can significantly improve the precision of cosmological parameters.