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

Maria Giovanna Dainotti, Avik Banerjee, Andre' LeClair, Giovanni Montani

Abstract

In view of the current and increasing evidence of a running Hubble constant, we investigate its redshift dependence within the flat $Λ$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 $Ω_{m0}$, 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 $Λ$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.

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

Abstract

In view of the current and increasing evidence of a running Hubble constant, we investigate its redshift dependence within the flat 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 and , 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 (), Big Bang Nucleosynthesis (BBN, ), and inflationary scales (). 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 at finite , while the power-law form, , approaches zero asymptotically as . In future studies, independent high-redshift observations and extensions beyond CDM, such as modified gravity, could allow a comparative study of the two parameterizations beyond the SNe~Ia regime and their high- physical implications.
Paper Structure (14 sections, 82 equations, 4 figures, 3 tables)

This paper contains 14 sections, 82 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Flat $\Lambda CDM$ cosmological model
  4. The effective running Hubble constant
  5. Quantum vacuum energy density and its renormalization group properties as the origin of the Hubble tension
  6. Theoretical Realizations of the Power-Law Running Hubble Constant
  7. Logarithmic and Power-Law Parameterizations of the Hubble Constant
  8. Model Fitting and Statistical Criteria
  9. Equivalence condition for logarithmic and power-law parametrizations of the Hubble constant
  10. The Data Sample
  11. Data Analysis and Results
  12. Discussion of our results
  13. Discussion in relation to the literature
  14. Summary and Conclusions

Figures (4)

  • Figure 1: Compilation of Hubble constant ($H_0$) measurements from diverse cosmological probes reported in the literature (updated following 2025JHEAp..4800405D and 2025PDU....4901965D). Each point represents the central value of a measurement, with error bars indicating the quoted $1\sigma$ uncertainty. For measurements reporting separate statistical (stat) and systematic (sys) uncertainties, the thick inner bars represent the $1\sigma$ statistical errors, while the thin outer bars denote the total $1\sigma$ uncertainties obtained from the quadrature sum of the statistical and systematic components. References: [1] 2021ApJ...909..218A, [2] Gayathri:2020mra, [3] 2020arXiv200914199M, [4] 2021AA...646A..65M, [5] 2017Natur.551...85A, [6] 2024PhRvD.109f3508P, [7] 2024MNRAS.535..961B, [8] LIGOScientific:2021aug, [9] 2023MNRAS.518.2201D, [10] 2023MNRAS.518.2201D, [11] 2023ApJS..264...46L, [12] 2023ApJS..264...46L, [13] 2022MNRAS.516.4862J, [14] 2023ApJ...946L..49L, [15] 2025PDU....4801926K, [16] 2021MNRAS.502.2065D, [17] 2021MNRAS.502.2065D, [18] 2023AA...673A...9S, [19] 2021MNRAS.501..784D, [20] 2020AA...643A.165B, [21] 2020AA...643A.165B, [22] 2020MNRAS.497L..56Y, [23] 2020AA...639A.101M, [24] 2021MNRAS.501.1823B, [25] 2021MNRAS.503.2179Q, [26] 2020ApJ...895L..29L, [27] 2019ApJ...886L..23L, [28] 2025ApJ...979...13P, [29] 2020MNRAS.494.6072S, [30] 2020MNRAS.498.1420W, [31] 2019MNRAS.484.4726B, [32] 2017MNRAS.465.4914B, [33] 2024AA...682A..20C, [34] 2023MNRAS.519.4938Y, [35] 2023Sci...380.1322K, [36] 2018MNRAS.474.1250F, [37] 2025MNRAS.538.1264C, [38] 2025ApJ...979L...9S, [39] 2022MNRAS.514.4620D, [40] 2020MNRAS.496.3402D, [41] 2025AA...702A..41V, [42] 2024ApJ...962...60D, [43] 2021ApJ...911...65B, [44] 2021AA...647A..72K, [45] 2022MNRAS.511.6160K, [46] 2020ApJ...896....3K, [47] 2020AJ....160...71S, [48] 2024arXiv241208449S, [49] 2020ApJ...891L...1P, [50] 2020ApJ...889....5H, [51] 2024ApJ...963...83H, [52] 2023ApJ...954L..31S, [53] 2021ApJ...908L...5S, [54] 2020ApJ...891...57F, [55] 2019ApJ...886L..27R, [56] 2019ApJ...882...34F, [57] 2019ApJ...886...61Y, [58] 2017arXiv170201118J, [59] 2025arXiv250311769H, [60] 2022ApJ...932...15A, [61] 2023ApJ...954L..31S, [62] 2021ApJ...911...65B, [63] 2025ApJ...987...87J, [64] 2022ApJ...938...36R, [65] 2021ApJ...908L...6R, [66] 2020AA...643A.115B, [67] 2019ApJ...876...85R, [68] 2020PhRvR...2a3028C, [69] 2018ApJ...869...56B, [70] 2018AA...609A..72D, [71] 2018MNRAS.477.4534F, [72] 2018MNRAS.476.3861F, [73] 2016ApJ...826...56R, [74] 2017JCAP...03..056C, [75] 2012ApJ...758...24F, [76] 2021ApJ...911...65B, [77] 2022ApJ...935...83K, [78] 2024ApJ...977..120R, [79] 2024ApJ...977..120R, [80] 2024ApJ...966...20L, [81] 2025ApJ...985..203F, [82] 2025ApJ...988...97L, [83] 2020JCAP...05..005D, [84] 2020JCAP...06..001C, [85] 2020JCAP...05..032P, [86] 2020JCAP...05..042I, [87] 2017MNRAS.470.2617A, [88] 2025JCAP...02..021A, [89] 2022JCAP...11..039S, [90] 2022JCAP...11..039S, [91] 2021PhRvD.103b3538P, [92] 2021PhRvD.104b2003D, [93] 2025PhRvD.111h3534G, [94] 2020JCAP...12..047A, [95] 2025JCAP...11..062L, [96] 2020JCAP...12..047A, [97] 2019CoTPh..71..826Z, [98] 2013ApJS..208...19H, [99] 2025JCAP...11..062L, [100] 2021PhRvD.104h3509B, [101] 2020ApJ...904L..17P, [102] 2020AA...641A...6P, [103] 2020AA...641A...6P, [104] 2016AA...594A..13P, and [105] 2025JCAP...11..062L.
  • Figure 2: Joint two-dimensional posterior distributions of the cosmological parameters $H_{0}$ and $\Omega_{\mathrm{m}0}$ obtained from the 20 redshift-bin analysis within the $\Lambda$CDM model. Red contours show results for the Master Sample including low-$z$ supernovae, and light-seagreen contours correspond to the sample without low-$z$ data. The inner and outer contours denote the $1\sigma$ (68%) and $2\sigma$ (95%) confidence regions, respectively.
  • Figure 3: Corner plots showing the posterior distributions of the model parameters obtained from the MCMC analysis of the Master SNe Ia Sample. Figures (a) and (b) correspond to the logarithmic parameterization of the Hubble constant, with free parameters $\tilde{H}_0^{\mathrm{Log}}$ and $\hat{b}$, using datasets including and excluding low-redshift SNe Ia, respectively. Figures (c) and (d) show the corresponding results for the power-law parameterization, characterized by the parameters $\tilde{H}_0^{\mathrm{PL}}$ and $\alpha$. The contours denote the $68\%$ and $95\%$ confidence regions, with the marginalized one-dimensional posterior distributions shown along the diagonal.
  • Figure 4: Best-fit model predictions for the Hubble constant reconstructed from the 20-bin analysis as a function of redshift, obtained from the Master SNe Ia Sample. Figures (a) and (b) correspond to the logarithmic parameterization, $\mathcal{H}_0^{\mathrm{Log}}(z)$, and the power-law parameterization, $\mathcal{H}_0^{\mathrm{PL}}(z)$, using datasets including and excluding low-redshift SNe Ia, respectively. The black points represent the binned observational measurements, $H_0^{\mathrm{obs}}$, with associated uncertainties. The blue dashed line shows the best-fit prediction of the logarithmic model, while the red solid line shows the best-fit prediction of the power-law model. The quantity $\langle z \rangle$ denotes the mean redshift in each bin obtained from the 20-bin analysis.