Parameterizations of the Hubble Constant from the Binned Type Ia Supernova Master Sample: Logarithmic versus Power-law Forms

Abstract

Motivated by the Hubble tension and the increasing debate about the redshift-dependency in the inferred Hubble constant, we investigate its dependence within the flat $\Lambda$CDM framework using a 20-bin analysis of the Master Supernovae type Ia (SNe Ia) Sample, considering cases with and without very low-redshift data. The main quantity studied is $H_0(z)/H_0$, where $H_0$ is the Hubble constant today, and $H_0(z)$ is the binned data-driven estimate of the value of $H_0$ inferred from the SNe Ia data within redshift intervals at $z>0$, such that $H_0(0)=H_0$. From the binned analyses, we obtain best-fitting values of $H_0$ and ${\Omega }_{m0}$, and employ logarithmic and power-law parameterizations, which are statistically consistent within uncertainties over the redshift range considered. 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, though these two regimes lie beyond the direct observational constraints. A distinct asymptotic behavior emerges at very-high redshift: the logarithmic form exhibits a vanishing behaviour of $H_0$ at finite $z$, while the power-law form approaches zero asymptotically as $z\to \infty$. In future studies, independent high-redshift observations and extensions beyond $\Lambda$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.

28 pages, 4 figures, Accepted for publication in ApJ