Constraints on transition redshift utilizing the latest H(z) measurements and comments on the Hubble tension

The motivation of this paper is to obtain reliable constraints of transition redshift ($z_{ztr}$) and, in combination with the evolution of the Hubble constant ($H_{0}$) that could alleviate the Hubble tension, discuss the possible origin of the tension. Utilizing the latest H(z) measurements and different methods ($\Lambda$CDM model, Cosmography, and Gaussian process method), we investigated the impact of methodology and dataset on $z_{ztr}$ constraints, and find that the choice of method has a greater impact on $z_{tr}$ than the observations themselves. Through a statistical analysis of the $z_{ztr}$ constraints from 2004 to 2024, we find that total $z_{tr}$ constraints (2004$-$2024) can be well described by a Gaussian function with the mean value 0.65 and the standard deviation 0.16; that is, $\bar{z}_{tr}$(all) = 0.65 $\pm$ 0.16. And we confirmed that both dataset and methodology can indeed significantly affect the final constraints. The screened $z_{tr}$ constraints with free $H_{0}$ gives a new result $\bar{z}_{tr}$(free) = 0.64 $\pm$ 0.16. Coincidentally, the $z_{tr}$ results overlap with the initial moment of $H_{0}$ evolution ($H_{0}$ value starts to deviate from the Planck result). This may suggest that the Hubble tension might be closely related to this particular period in the evolution of the Universe.