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

This paper investigates the redshift dependence of the Hubble constant using binned Type Ia supernova data within the flat $\Lambda$CDM framework, comparing logarithmic and power-law parameterizations that agree at low redshifts but diverge significantly when extrapolated to early cosmic epochs like BBN and inflation, with the logarithmic form predicting a finite redshift where the Hubble parameter vanishes while the power-law form approaches zero asymptotically.

The Gist

Imagine you are trying to measure the speed of a car. You have two very different ways of doing it:

1. The Local Method: You stand on the side of the road with a radar gun and time the car as it zooms past you right now.
2. The Historical Method: You look at the car's old logbook from when it left the factory years ago and calculate how fast it must have been going to get here.

In cosmology, these are the two ways we measure the Hubble Constant ($H_0$), which is essentially the "speed limit" of the expanding universe.

• The Local Method (using nearby exploding stars called Supernovae) says the universe is expanding at about 73 units.
• The Historical Method (using the Cosmic Microwave Background, the "baby picture" of the universe) says it's expanding at about 67 units.

This mismatch is called the Hubble Tension. It's like the car's logbook saying it was doing 60 mph, but your radar gun says 73 mph. Something is wrong with the math, the car, or the road.

The Paper's Big Idea: The "Running" Speedometer

The authors of this paper propose a radical idea: Maybe the speedometer isn't broken; maybe the speed limit itself changes depending on how far back in time you look.

They suggest that the Hubble Constant isn't a single, fixed number. Instead, it's a "Running Hubble Constant"—a speed that evolves as the universe gets older (or younger, if you look back in time).

To test this, they took a massive dataset of 3,700+ exploding stars (Supernovae) and split them into 20 time-buckets (bins). They asked: Does the calculated expansion rate change as we look at stars from different eras?

The Two Competing Theories (The Analogies)

The paper compares two different mathematical "recipes" to describe how this speed changes. Think of them as two different ways to describe how a balloon inflates.

1. The Power-Law Recipe (The "Exponential Balloon")

• The Metaphor: Imagine a balloon that inflates faster and faster. The rate of inflation follows a strict power rule. As you go back in time (higher redshift), the speed drops off smoothly, like a curve sliding down a hill. It never hits a hard stop; it just gets closer and closer to zero forever.
• The Science: This is the "Power-Law" model. It suggests the universe's expansion rate changes gradually and predictably over time.

2. The Logarithmic Recipe (The "Speed Limit Sign")

• The Metaphor: Imagine a road where the speed limit changes based on a logarithmic scale. As you drive further back in time, the speed limit drops, but it hits a "hard floor" at a specific point. If you go too far back, the math says the speed limit would hit zero or even become impossible.
• The Science: This is the "Logarithmic" model. It suggests that at a certain point in the very early universe, the expansion rate might have behaved differently, perhaps avoiding the "Big Bang Singularity" (the moment where everything was crushed into a single point of infinite density).

What Did They Find?

1. At "Low" Redshifts (Recent History):
When they looked at the data from the last few billion years (the "low-z" data), both recipes gave the exact same answer.

• Analogy: It's like driving on a highway where both the "Exponential" and "Logarithmic" speed limit signs look identical for the first 10 miles. You can't tell them apart.
• Result: Both models fit the data perfectly and suggest the expansion rate is slightly higher today than it was in the past.

2. At "High" Redshifts (The Deep Past):
This is where the rubber meets the road. The authors took their best-fit equations and extrapolated them (projected them forward) to the very beginning of the universe:

• The CMB Era (380,000 years after the Big Bang): Both models still agreed closely.
• The BBN Era (3 minutes after the Big Bang): The models started to drift apart by about 5–10%.
• The Inflationary Era (The very first split-second): The models diverged wildly.
  • The Power-Law model says the speed just gets smaller and smaller as you go back, approaching zero but never stopping.
  • The Logarithmic model says the speed hits a "wall" at a specific time. It suggests the universe might have had a minimum size and never actually reached a "singularity" (a point of infinite density). It implies the universe might have bounced or existed in a state before the Big Bang.

Why Does This Matter?

The paper concludes that while we can't tell the difference between these two theories with our current telescopes (because we can't see far enough back yet), they are fundamentally different stories about the beginning of the universe.

• If the Power-Law is right, the Big Bang was a true singularity, and the universe started from nothing.
• If the Logarithmic model is right, the Big Bang might have been a "bounce" from a previous state, and the laws of physics (specifically gravity) might have changed slightly in the early universe.

The Takeaway

The authors are essentially saying: "We have a mystery (the Hubble Tension). We have two different theories that both explain the data we have right now. They look identical in our backyard, but they tell completely different stories about the universe's infancy. To solve the mystery, we need to build better telescopes to look further back in time and see which story is actually true."

It's a reminder that the universe might be more dynamic and complex than our simple "standard model" suggests, and the speed limit of the cosmos might have been different when the universe was a toddler than it is today.

Technical Summary

Here is a detailed technical summary of the paper "Parameterizations of the Hubble Constant: Logarithmic vs Power-Law Expansion from the Binned Master Sample of SNe Ia" by Dainotti et al.

1. Problem Statement

The paper addresses the persistent Hubble tension, a significant discrepancy between the local measurement of the Hubble constant ($H_0$) derived from Type Ia Supernovae (SNe Ia) and the Cosmic Distance Ladder ($H_0 \approx 73.04 \pm 1.04$ km s$^{-1}$ Mpc$^{-1}$) versus the value inferred from early-Universe Cosmic Microwave Background (CMB) data under the standard $\Lambda$CDM model ($H_0 \approx 67.4 \pm 0.5$ km s$^{-1}$ Mpc$^{-1}$).

The authors investigate whether this tension arises from a running Hubble constant, where the effective expansion rate $H_0(z)$ varies with redshift, rather than being a constant. They aim to test two specific phenomenological parameterizations of this redshift dependence:

1. Logarithmic Form: Motivated by quantum vacuum energy renormalization group (RG) flow and the gravitational Casimir effect (LeClair, 2026).
2. Power-Law Form: Motivated by modified gravity scenarios (specifically $f(R)$ gravity) and scaling laws in astrophysical sources (Dainotti et al., 2021, 2025).

2. Methodology

Data Source:
The study utilizes the Master SNe Ia Sample (Dainotti et al., 2025), a compilation of 3,714 unique SNe Ia derived from four major catalogs: DES, Pantheon+, Pantheon, and JLA.

• Two Configurations: The analysis is performed on two subsets:
  1. With Low-z: Full sample ($0.00122 \le z_{HD} \le 2.26137$).
  2. Without Low-z: Excluding very low-redshift supernovae ($0.01006 \le z_{HD} \le 2.26137$) to mitigate peculiar velocity effects.
• Binning: The data is divided into 20 equi-populated redshift bins.

Statistical Framework:

• Likelihood: The authors employ a non-Gaussian likelihood approach, acknowledging that residuals of distance moduli normalized by the full covariance matrix are non-Gaussian. This reduces parameter uncertainties by up to ~43% compared to standard Gaussian assumptions.
• Parameter Estimation: Markov Chain Monte Carlo (MCMC) sampling (via the Cobaya framework) is used to fit the cosmological parameters ($H_0$ and $\Omega_{m0}$) within the flat $\Lambda$CDM framework for each bin.
• Model Fitting: The binned $H_0$ values are fitted to two functional forms:
  • Logarithmic: $H_0^{\text{Log}}(z) = \tilde{H}_0^{\text{Log}} \sqrt{1 - \hat{b} \ln(1+z)}$
  • Power-Law: $H_0^{\text{PL}}(z) = \tilde{H}_0^{\text{PL}} (1+z)^{-\alpha}$
• Model Comparison: Goodness-of-fit is assessed using reduced chi-squared ($\chi^2_{\text{red}}$) and the Bayesian Information Criterion (BIC).

3. Key Contributions

1. Equivalence at Low Redshift: The paper rigorously demonstrates that the logarithmic and power-law parameterizations are mathematically equivalent in the low-redshift regime ($z \ll 1$) under the condition $\hat{b} = 2\alpha$. The Taylor expansions of both functions converge to the same linear term in $\ln(1+z)$.
2. High-Redshift Extrapolation: The authors extrapolate the best-fit models to epochs far beyond the observational data, specifically:
   • CMB era ($z \approx 1100$)
   • Big Bang Nucleosynthesis (BBN, $z \sim 10^9$)
   • Inflationary era ($z \sim 10^{20}$)
3. Physical Distinction: The study identifies a qualitative divergence at very high redshifts:
   • The Logarithmic model predicts a vanishing $H_0(z)$ at a finite maximum redshift ($z_{\text{max}}$), implying a minimum scale factor ($a_{\text{min}}$) and potentially avoiding the Big Bang singularity.
   • The Power-Law model approaches zero asymptotically as $z \to \infty$, consistent with standard Big Bang cosmology containing a singularity.

4. Key Results

Best-Fit Parameters (Table 2):

• With Low-z Data:
  • Logarithmic: $\tilde{H}_0^{\text{Log}} = 69.909 \pm 0.096$, $\hat{b} = 0.023 \pm 0.013$.
  • Power-Law: $\tilde{H}_0^{\text{PL}} = 69.909 \pm 0.096$, $\alpha = 0.012 \pm 0.006$.
  • Observation: The condition $\hat{b} \approx 2\alpha$ holds ($0.023 \approx 2 \times 0.012$), confirming equivalence.
• Without Low-z Data:
  • Logarithmic: $\tilde{H}_0^{\text{Log}} = 69.839 \pm 0.103$, $\hat{b} = 0.017 \pm 0.013$.
  • Power-Law: $\tilde{H}_0^{\text{PL}} = 69.839 \pm 0.104$, $\alpha = 0.009 \pm 0.006$.
  • Observation: The relation $\hat{b} \approx 2\alpha$ holds ($0.017 \approx 2 \times 0.009$).

Model Consistency:

• Both models provide statistically indistinguishable fits to the current SNe Ia data ($\chi^2_{\text{red}} \approx 1.24$ with low-z; $\approx 2.08$ without low-z).
• The inclusion of low-z data yields a slightly better fit quality but does not alter the fundamental conclusion of equivalence.

High-Redshift Extrapolation (Table 3):

• CMB ($z=1100$): The models differ by only $\sim 0.37\%$ (without low-z) to $\sim 0.75\%$ (with low-z).
• BBN ($z=10^9$): The discrepancy grows to $\sim 4.17\%$ (without low-z) and $\sim 9.32\%$ (with low-z).
• Inflation ($z=10^{20}$):
  • Power-Law: $H_0^{\text{PL}}$ remains finite and non-zero (e.g., $\sim 40$ km s$^{-1}$ Mpc$^{-1}$ for the "without low-z" case).
  • Logarithmic: $H_0^{\text{Log}}$ vanishes at a finite $z_{\text{max}}$.
    • With low-z: $z_{\text{max}} \approx 3.33 \times 10^{18}$.
    • Without low-z: $z_{\text{max}} \approx 2.59 \times 10^{25}$.
    • Beyond $z_{\text{max}}$, the model is undefined, suggesting a "bounce" or a pre-Big Bang phase where the scale factor never reaches zero.

5. Significance and Implications

• Resolution of Hubble Tension: The systematic decrease of $H_0(z)$ with increasing redshift (found in both parameterizations) suggests that the "local" $H_0$ is higher than the "early-Universe" $H_0$ because the expansion rate was slower in the past. This running behavior offers a phenomenological pathway to reconcile the tension without necessarily invoking new physics in the dark energy sector, though it implies deviations from strict $\Lambda$CDM.
• Singularity Avoidance: The logarithmic parameterization, rooted in RG flow of vacuum energy, offers a theoretical mechanism to avoid the initial Big Bang singularity by introducing a minimum scale factor. This provides a testable distinction between standard cosmology and quantum-gravity-inspired models.
• Future Directions: The paper concludes that while the two models are degenerate at low redshifts, future observations at high redshifts ($z > 2$) using Gamma-Ray Bursts (GRBs), Quasars, and strong gravitational lensing are essential to break the degeneracy. Specifically, detecting whether $H_0(z)$ vanishes at a finite redshift or asymptotically approaches zero will distinguish between the logarithmic (RG/vacuum energy) and power-law (modified gravity) origins of the running Hubble constant.

In summary, this work provides a robust statistical validation that a running Hubble constant is a viable description of SNe Ia data, establishes the mathematical equivalence of two leading theoretical models at low redshift, and highlights their divergent cosmological implications in the early Universe.