Parameterizations of the Hubble Constant: Logarithmic vs Power-Law Expansion from the Binned Master Sample of SNe Ia

In view of the current and increasing evidence of a running Hubble constant, we investigate its redshift dependence within the flat $\mathrm{\Lambda }$CDM framework using a 20-bin analysis of the Master SNe~Ia Sample \citep{2025JHEAp…4800405D}, considering cases with and without very low-redshift data. For each case, we obtain best-fitting values of $H_0$ and ${\mathrm{\Omega }}_{m0}$, and employ both logarithmic \citep{2025arXiv250902636L} and power-law \citep{2021ApJ…912…150D,2022Galax…10…24D,2025JHEAp…4800405D} parameterizations. The two parameterizations are consistent over the redshift range considered and coincide for low redshifts. To assess their behavior at earlier epochs, we extrapolate both forms to the Cosmic Microwave Background radiation (CMB) era ($z\simeq 1100$), Big Bang Nucleosynthesis (BBN, $z\sim {10}^9$), and inflationary scales ($z\sim {10}^{20}$). The reconstructed Hubble constant remains nearly indistinguishable up to the CMB scale, diverges at the few-to-ten percent level around BBN, and differs more substantially when extrapolated to inflationary redshifts. A qualitative distinction emerges at very-high redshift: the logarithmic form predicts a vanishing of ${\mathcal{H}}_0^{\mathrm{L}\mathrm{o}\mathrm{g}}(z)$ at finite $z$, while the power-law form, ${\mathcal{H}}_0^{\mathrm{P}\mathrm{L}}(z)$, approaches zero asymptotically as $z\to \mathrm{\infty }$. In future studies, independent high-redshift observations and extensions beyond $\mathrm{\Lambda }$CDM, such as $f(R)$ modified gravity, could allow a comparative study of the two parameterizations beyond the SNe~Ia regime and their high-$z$ physical implications.