Model-independent test of the running Hubble constant from the Type Ia supernovae and the Hubble parameter data

In this study, we model-independently investigate the behaviour of running Hubble constant, characterized by the fit function $H_{\rm 0}(z)=\tilde{H_{\rm {0}}}/(1+z)<^>{\alpha }$, where alpha represents the evolutionary parameter and ${\tilde{H_{\rm {0}}}}$ corresponds to the current value of Hubble constant. Our analysis utilizes the expansion rate E(z) data points measured from the Pantheon + Multi-Cycle Treasury compilation of Type Ia supernova data, the measurements of H-0 obtained by Riess et al., and the Hubble parameter H(z) data obtained from the differential ages of passive galaxies [known as cosmic chronometer (CC) method] and from the baryon acoustic oscillation (BAO) in the radial direction of galaxy clustering. To resolve the redshift mismatch problem between the E(z) and H(z) data sets, we adopt the Hubble parameter data obtained via CC or BAO along with the measurements of H-0 obtained by Riess et al. to reconstruct the H(z) function using the Gaussian process. Our constraint yields alpha values of 0.125 +/- 0.063 or 0.095 +/- 0.052 when combining six pairs of the E(z) data and the reconstructed H(z) points via CC or BAO. These findings reveal that the Hubble constant may evolve with redshift, exhibiting a slowly decreasing trend, with alpha coefficients consistent with zero only at 2.0 sigma or 1.8 sigma. Therefore, the running Hubble constant might offer a promising resolution to the Hubble tension, and its reliability should be further tested through high-precision measurement at higher redshifts, such as the upcoming gamma-ray bursts and quasars.