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

TL;DR

This study compares logarithmic and power-law models of the Hubble constant's redshift dependence using supernova data, revealing their similarities at low redshift but divergence at very high redshift, with implications for cosmology.

Contribution

It introduces a detailed comparison of two parameterizations of the Hubble constant's evolution across a wide redshift range within the flat ΛCDM framework.

Findings

Both parameterizations agree at low redshift.

Differences emerge at high redshift, especially near inflationary scales.

Logarithmic form predicts a finite redshift where Hubble parameter vanishes.

Abstract

In view of the current and increasing evidence of a running Hubble constant, we investigate its redshift dependence within the flat $\Lambda$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 $\Omega_{m0}$, 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 ($z\simeq1100$), Big Bang Nucleosynthesis (BBN, $z\sim10^{9}$), and inflationary scales ($z\sim10^{20}$). 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 $\mathcal{H}_0^{\mathrm{Log}}(z)$ at finite $z$, while the power-law form, $\mathcal{H}_0^{\mathrm{PL}}(z)$, approaches zero asymptotically as $z \rightarrow \infty$. 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.